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L3 Lab 3

case pos R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j ·

specialize ih hp.2 hn HI

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ih : I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ t, I ≤ f i ⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI

rcases ih with (ih | ih | ⟨k, hkt, ih⟩)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ih : I ≤ f a ⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) ·

left

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inl.h R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ih : I ≤ f a ⊢ I ≤ f a

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left

exact ih

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inr.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ih : I ≤ f b ⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih ·

right

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inr.inl.h R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ih : I ≤ f b ⊢ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right

left

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inr.inl.h.h R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ih : I ≤ f b ⊢ I ≤ f b

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left

exact ih

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inr.inr.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) k : ι hkt : k ∈ t ih : I ≤ f k ⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih ·

right

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inr.inr.intro.intro.h R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) k : ι hkt : k ∈ t ih : I ≤ f k ⊢ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right

right

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.inr.inr.intro.intro.h.h R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) k : ι hkt : k ∈ t ih : I ≤ f k ⊢ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right

exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩

exfalso

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso

rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩

rw [Finset.coe_insert, Set.biUnion_insert] at h

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h

have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs)

rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs)

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inl.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i ha : r + s ∈ ↑(f a) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) ·

exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inl.inr R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i hb : r + s ∈ ↑(f b) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) ·

exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inr.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i hi : r + s ∈ ↑(f i) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) ·

exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inr.inr R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i ht : r + s ∈ ⋃ x ∈ ↑t, ↑(f x) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) ·

rw [Set.mem_iUnion₂] at ht

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) ·

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inr.inr R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i ht : ∃ i, ∃ (_ : i ∈ ↑t), r + s ∈ ↑(f i) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht

rcases ht with ⟨j, hjt, hj⟩

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inr.inr.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hr : r ∈ ↑(Finset.inf t f) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i j : ι hjt : j ∈ ↑t hj : r + s ∈ ↑(f j) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩

simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.h.intro.intro.inr.inr.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R I : Ideal R n : ℕ ih : ∀ {s : Finset ι} {a b : ι}, (∀ i ∈ s, IsPrime (f i)) → Finset.card s = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i a b i : ι t : Finset ι hit : i ∉ t hn : Finset.card t = n h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ (↑(f i) ∪ ⋃ x ∈ ↑t, ↑(f x)) hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) Ht : ¬∃ j ∈ t, f j ≤ f i Ha : ¬f a ≤ f i Hb : ¬f b ≤ f i Hi : ¬I ≤ f i this : ¬I ⊓ f a ⊓ f b ⊓ Finset.inf t f ≤ f i r : R hri : r ∉ ↑(f i) hrb : r ∈ ↑(f b) hrI : r ∈ ↑I hra : r ∈ ↑(f a) HI : ¬↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) s : R hsI : s ∈ ↑I hs : s ∉ ↑(f a) ∪ ↑(f b) ∪ ⋃ j ∈ ↑t, ↑(f j) hsi : s ∈ f i j : ι hjt : j ∈ ↑t hj : r + s ∈ ↑(f j) hr : ∀ i ∈ t, r ∈ f i ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr

exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt)

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr

Mathlib.RingTheory.Ideal.Operations.1127_0.5qK551sG47yBciY

theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by

classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical

by_cases has : a ∈ s

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∈ s ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s ·

obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t ⊢ ∃ i ∈ insert a t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩

by_cases hbt : b ∈ t

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∈ t ⊢ ∃ i ∈ insert a t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t ·

obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a (insert b u)), ↑(f i) has : a ∈ insert a (insert b u) hbt : b ∈ insert b u ⊢ ∃ i ∈ insert a (insert b u), I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩

have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a (insert b u)), ↑(f i) has : a ∈ insert a (insert b u) hbt : b ∈ insert b u ⊢ ∀ i ∈ u, IsPrime (f i)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by

intro i hiu

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a (insert b u)), ↑(f i) has : a ∈ insert a (insert b u) hbt : b ∈ insert b u i : ι hiu : i ∈ u ⊢ IsPrime (f i)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu

refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a (insert b u)), ↑(f i) has : a ∈ insert a (insert b u) hbt : b ∈ insert b u i : ι hiu : i ∈ u ⊢ i ≠ a

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a (insert b u)), ↑(f i) has : a ∈ insert a (insert b u) hbt : b ∈ insert b u i : ι hiu : i ∈ u ⊢ i ≠ b

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R b : ι I : Ideal R u : Finset ι hbu : b ∉ u hbt : b ∈ insert b u i : ι hiu : i ∈ u hat : i ∉ insert b u hp : ∀ i_1 ∈ insert i (insert b u), i_1 ≠ i → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i (insert b u)), ↑(f i_1) has : i ∈ insert i (insert b u) ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a : ι I : Ideal R u : Finset ι i : ι hiu : i ∈ u hbu : i ∉ u hat : a ∉ insert i u hp : ∀ i_1 ∈ insert a (insert i u), i_1 ≠ a → i_1 ≠ i → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert a (insert i u)), ↑(f i_1) has : a ∈ insert a (insert i u) hbt : i ∈ insert i u ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a (insert b u)), ↑(f i) has : a ∈ insert a (insert b u) hbt : b ∈ insert b u hp' : ∀ i ∈ u, IsPrime (f i) ⊢ ∃ i ∈ insert a (insert b u), I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R u : Finset ι hbu : b ∉ u hat : a ∉ insert b u hp : ∀ i ∈ insert a (insert b u), i ≠ a → i ≠ b → IsPrime (f i) h : I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ u, I ≤ f i has : a ∈ insert a (insert b u) hbt : b ∈ insert b u hp' : ∀ i ∈ u, IsPrime (f i) ⊢ ∃ i ∈ insert a (insert b u), I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h

rwa [Finset.exists_mem_insert, Finset.exists_mem_insert]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∉ t ⊢ ∃ i ∈ insert a t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] ·

have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∉ t ⊢ ∀ j ∈ t, IsPrime (f j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by

intro j hj

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∉ t j : ι hj : j ∈ t ⊢ IsPrime (f j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj

refine' hp j (Finset.mem_insert_of_mem hj) _ _

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∉ t j : ι hj : j ∈ t ⊢ j ≠ a

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∉ t j : ι hj : j ∈ t ⊢ j ≠ b

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R b : ι I : Ideal R t : Finset ι hbt : b ∉ t j : ι hj : j ∈ t hat : j ∉ t hp : ∀ i ∈ insert j t, i ≠ j → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert j t), ↑(f i) has : j ∈ insert j t ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a : ι I : Ideal R t : Finset ι hat : a ∉ t h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t j : ι hj : j ∈ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ j → IsPrime (f i) hbt : j ∉ t ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i) has : a ∈ insert a t hbt : b ∉ t hp' : ∀ j ∈ t, IsPrime (f j) ⊢ ∃ i ∈ insert a t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hat : a ∉ t hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → IsPrime (f i) h : I ≤ f a ∨ ∃ i ∈ t, I ≤ f i has : a ∈ insert a t hbt : b ∉ t hp' : ∀ j ∈ t, IsPrime (f j) ⊢ ∃ i ∈ insert a t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

rwa [Finset.exists_mem_insert]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∉ s ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] ·

by_cases hbs : b ∈ s

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∉ s hbs : b ∈ s ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s ·

obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) has : a ∉ insert b t hbs : b ∈ insert b t ⊢ ∃ i ∈ insert b t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩

have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) has : a ∉ insert b t hbs : b ∈ insert b t ⊢ ∀ j ∈ t, IsPrime (f j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by

intro j hj

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) has : a ∉ insert b t hbs : b ∈ insert b t j : ι hj : j ∈ t ⊢ IsPrime (f j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj

refine' hp j (Finset.mem_insert_of_mem hj) _ _

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) has : a ∉ insert b t hbs : b ∈ insert b t j : ι hj : j ∈ t ⊢ j ≠ a

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) has : a ∉ insert b t hbs : b ∈ insert b t j : ι hj : j ∈ t ⊢ j ≠ b

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R b : ι I : Ideal R t : Finset ι hbt : b ∉ t h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) hbs : b ∈ insert b t j : ι hj : j ∈ t hp : ∀ i ∈ insert b t, i ≠ j → i ≠ b → IsPrime (f i) has : j ∉ insert b t ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a : ι I : Ideal R t : Finset ι j : ι hj : j ∈ t hbt : j ∉ t hp : ∀ i ∈ insert j t, i ≠ a → i ≠ j → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert j t), ↑(f i) has : a ∉ insert j t hbs : j ∈ insert j t ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i) has : a ∉ insert b t hbs : b ∈ insert b t hp' : ∀ j ∈ t, IsPrime (f j) ⊢ ∃ i ∈ insert b t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case pos.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R t : Finset ι hbt : b ∉ t hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → IsPrime (f i) h : I ≤ f b ∨ ∃ i ∈ t, I ≤ f i has : a ∉ insert b t hbs : b ∈ insert b t hp' : ∀ j ∈ t, IsPrime (f j) ⊢ ∃ i ∈ insert b t, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

rwa [Finset.exists_mem_insert]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∉ s hbs : b ∉ s ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert]

rcases s.eq_empty_or_nonempty with hse | hsne

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert]

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∉ s hbs : b ∉ s hse : s = ∅ ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne ·

subst hse

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R hp : ∀ i ∈ ∅, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I ⊆ ⋃ i ∈ ↑∅, ↑(f i) has : a ∉ ∅ hbs : b ∉ ∅ ⊢ ∃ i ∈ ∅, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse

rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R hp : ∀ i ∈ ∅, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I = ∅ has : a ∉ ∅ hbs : b ∉ ∅ ⊢ ∃ i ∈ ∅, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h

have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem)

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inl R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R hp : ∀ i ∈ ∅, i ≠ a → i ≠ b → IsPrime (f i) h : ↑I = ∅ has : a ∉ ∅ hbs : b ∉ ∅ this : ↑I ≠ ∅ ⊢ ∃ i ∈ ∅, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem)

exact absurd h this

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem)

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inr R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∉ s hbs : b ∉ s hsne : Finset.Nonempty s ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this ·

cases' hsne.bex with i his

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this ·

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inr.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R h : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) has : a ∉ s hbs : b ∉ s hsne : Finset.Nonempty s i : ι his : i ∈ s ⊢ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his

obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inr.intro.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t ⊢ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩

have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t ⊢ ∀ j ∈ t, IsPrime (f j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by

intro j hj

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t j : ι hj : j ∈ t ⊢ IsPrime (f j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj

refine' hp j (Finset.mem_insert_of_mem hj) _ _

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t j : ι hj : j ∈ t ⊢ j ≠ a

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t j : ι hj : j ∈ t ⊢ j ≠ b

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

rintro rfl

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t j : ι hj : j ∈ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ j → i_1 ≠ b → IsPrime (f i_1) has : j ∉ insert i t ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t j : ι hj : j ∈ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ j → IsPrime (f i_1) hbs : j ∉ insert i t ⊢ False

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

solve_by_elim only [Finset.mem_insert_of_mem, *]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inr.intro.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1) has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t hp' : ∀ j ∈ t, IsPrime (f j) ⊢ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

case neg.inr.intro.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R f : ι → Ideal R a b : ι I : Ideal R i : ι t : Finset ι left✝ : i ∉ t hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → IsPrime (f i_1) h : I ≤ f i ∨ ∃ i ∈ t, I ≤ f i has : a ∉ insert i t hbs : b ∉ insert i t hsne : Finset.Nonempty (insert i t) his : i ∈ insert i t hp' : ∀ j ∈ t, IsPrime (f j) ⊢ ∃ i_1 ∈ insert i t, I ≤ f i_1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

rwa [Finset.exists_mem_insert]

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R this : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) → ∃ i ∈ s, I ≤ f i ⊢ ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) ↔ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by

have aux := fun h => (bex_def.2 <| this h)

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R this : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) → ∃ i ∈ s, I ≤ f i aux : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) → ∃ x, ∃ (_ : x ∈ s), I ≤ f x ⊢ ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) ↔ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h)

simp_rw [exists_prop] at aux

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h)

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R this aux : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) → ∃ x ∈ s, I ≤ f x ⊢ ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) ↔ ∃ i ∈ s, I ≤ f i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux

refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u inst✝ : CommRing R s : Finset ι f : ι → Ideal R a b : ι hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i) I : Ideal R this aux : ↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) → ∃ x ∈ s, I ≤ f x x✝ : ∃ i ∈ s, I ≤ f i i : ι his : i ∈ s hi : I ≤ f i ⊢ ↑(f i) ⊆ ⋃ i ∈ ↑s, ↑(f i)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩

apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his)

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩

Mathlib.RingTheory.Ideal.Operations.1232_0.5qK551sG47yBciY

/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J K L I : Ideal R h : I = ⊤ ⊢ ⊤ = I * ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by

rw [mul_top, h]

theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by

Mathlib.RingTheory.Ideal.Operations.1301_0.5qK551sG47yBciY

theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R u : (Ideal R)ˣ ⊢ ↑u = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by

rw [isUnit_iff.mp u.isUnit, one_eq_top]

instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by

Mathlib.RingTheory.Ideal.Operations.1307_0.5qK551sG47yBciY

instance uniqueUnits : Unique (Ideal R)ˣ where default

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L I : Ideal S x y : R hx : x ∈ ⇑f ⁻¹' ↑I hy : y ∈ ⇑f ⁻¹' ↑I ⊢ x + y ∈ ⇑f ⁻¹' ↑I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by

simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by

Mathlib.RingTheory.Ideal.Operations.1336_0.5qK551sG47yBciY

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L I : Ideal S x y : R hx : f x ∈ I hy : f y ∈ I ⊢ f x + f y ∈ I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢

exact add_mem hx hy

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢

Mathlib.RingTheory.Ideal.Operations.1336_0.5qK551sG47yBciY

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L I : Ideal S ⊢ 0 ∈ { carrier := ⇑f ⁻¹' ↑I, add_mem' := (_ : ∀ {x y : R}, x ∈ ⇑f ⁻¹' ↑I → y ∈ ⇑f ⁻¹' ↑I → x + y ∈ ⇑f ⁻¹' ↑I) }.carrier

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by

simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem]

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by

Mathlib.RingTheory.Ideal.Operations.1336_0.5qK551sG47yBciY

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L I : Ideal S c x : R hx : x ∈ { toAddSubsemigroup := { carrier := ⇑f ⁻¹' ↑I, add_mem' := (_ : ∀ {x y : R}, x ∈ ⇑f ⁻¹' ↑I → y ∈ ⇑f ⁻¹' ↑I → x + y ∈ ⇑f ⁻¹' ↑I) }, zero_mem' := (_ : 0 ∈ ⇑f ⁻¹' ↑I) }.toAddSubsemigroup.carrier ⊢ c • x ∈ { toAddSubsemigroup := { carrier := ⇑f ⁻¹' ↑I, add_mem' := (_ : ∀ {x y : R}, x ∈ ⇑f ⁻¹' ↑I → y ∈ ⇑f ⁻¹' ↑I → x + y ∈ ⇑f ⁻¹' ↑I) }, zero_mem' := (_ : 0 ∈ ⇑f ⁻¹' ↑I) }.toAddSubsemigroup.carrier

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by

simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at *

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by

Mathlib.RingTheory.Ideal.Operations.1336_0.5qK551sG47yBciY

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L I : Ideal S c x : R hx : f x ∈ I ⊢ f c * f x ∈ I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at *

exact mul_mem_left I _ hx

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at *

Mathlib.RingTheory.Ideal.Operations.1336_0.5qK551sG47yBciY

/-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I J : Ideal R K L : Ideal S hK : K ≠ ⊤ ⊢ 1 ∉ comap f K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by

rw [mem_comap, map_one]

theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by

Mathlib.RingTheory.Ideal.Operations.1381_0.5qK551sG47yBciY

theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I J : Ideal R K L : Ideal S hK : K ≠ ⊤ ⊢ 1 ∉ K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one];

exact (ne_top_iff_one _).1 hK

theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one];

Mathlib.RingTheory.Ideal.Operations.1381_0.5qK551sG47yBciY

theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R g : G I : Ideal R hf : Set.LeftInvOn ⇑g ⇑f ↑I ⊢ map f I ≤ comap g I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by

refine' Ideal.span_le.2 _

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by

Mathlib.RingTheory.Ideal.Operations.1387_0.5qK551sG47yBciY

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R g : G I : Ideal R hf : Set.LeftInvOn ⇑g ⇑f ↑I ⊢ ⇑f '' ↑I ⊆ ↑(comap g I)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _

rintro x ⟨x, hx, rfl⟩

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _

Mathlib.RingTheory.Ideal.Operations.1387_0.5qK551sG47yBciY

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g

Mathlib_RingTheory_Ideal_Operations

case intro.intro R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R g : G I : Ideal R hf : Set.LeftInvOn ⇑g ⇑f ↑I x : R hx : x ∈ ↑I ⊢ f x ∈ ↑(comap g I)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩

rw [SetLike.mem_coe, mem_comap, hf hx]

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩

Mathlib.RingTheory.Ideal.Operations.1387_0.5qK551sG47yBciY

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g

Mathlib_RingTheory_Ideal_Operations

case intro.intro R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R g : G I : Ideal R hf : Set.LeftInvOn ⇑g ⇑f ↑I x : R hx : x ∈ ↑I ⊢ x ∈ I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx]

exact hx

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx]

Mathlib.RingTheory.Ideal.Operations.1387_0.5qK551sG47yBciY

theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R hK : IsPrime K x y : R ⊢ x * y ∈ Ideal.comap f K → x ∈ Ideal.comap f K ∨ y ∈ Ideal.comap f K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by

simp only [mem_comap, map_mul]

instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by

Mathlib.RingTheory.Ideal.Operations.1411_0.5qK551sG47yBciY

instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R hK : IsPrime K x y : R ⊢ f x * f y ∈ K → f x ∈ K ∨ f y ∈ K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul];

apply hK.2

instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul];

Mathlib.RingTheory.Ideal.Operations.1411_0.5qK551sG47yBciY

instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f✝ : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R f : F s : Set R ⊢ map f (span s) = span (⇑f '' s)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by

refine (Submodule.span_eq_of_le _ ?_ ?_).symm

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by

Mathlib.RingTheory.Ideal.Operations.1446_0.5qK551sG47yBciY

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s)

Mathlib_RingTheory_Ideal_Operations

case refine_1 R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f✝ : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R f : F s : Set R ⊢ ⇑f '' s ⊆ ↑(map f (span s))

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm ·

rintro _ ⟨x, hx, rfl⟩

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm ·

Mathlib.RingTheory.Ideal.Operations.1446_0.5qK551sG47yBciY

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s)

Mathlib_RingTheory_Ideal_Operations

case refine_1.intro.intro R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f✝ : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R f : F s : Set R x : R hx : x ∈ s ⊢ f x ∈ ↑(map f (span s))

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩;

exact mem_map_of_mem f (subset_span hx)

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩;

Mathlib.RingTheory.Ideal.Operations.1446_0.5qK551sG47yBciY

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s)

Mathlib_RingTheory_Ideal_Operations

case refine_2 R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f✝ : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R f : F s : Set R ⊢ map f (span s) ≤ Submodule.span S (⇑f '' s)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) ·

rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff]

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) ·

Mathlib.RingTheory.Ideal.Operations.1446_0.5qK551sG47yBciY

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s)

Mathlib_RingTheory_Ideal_Operations

case refine_2 R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f✝ : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R f : F s : Set R ⊢ ⇑f '' s ⊆ ↑(Submodule.span S (⇑f '' s))

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) · rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff]

exact subset_span

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) · rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff]

Mathlib.RingTheory.Ideal.Operations.1446_0.5qK551sG47yBciY

theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s)

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I✝ J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R I : Ideal S h : I = ⊤ ⊢ comap f I = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) · rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff] exact subset_span #align ideal.map_span Ideal.map_span variable {f I J K L} theorem map_le_of_le_comap : I ≤ K.comap f → I.map f ≤ K := (gc_map_comap f).l_le #align ideal.map_le_of_le_comap Ideal.map_le_of_le_comap theorem le_comap_of_map_le : I.map f ≤ K → I ≤ K.comap f := (gc_map_comap f).le_u #align ideal.le_comap_of_map_le Ideal.le_comap_of_map_le theorem le_comap_map : I ≤ (I.map f).comap f := (gc_map_comap f).le_u_l _ #align ideal.le_comap_map Ideal.le_comap_map theorem map_comap_le : (K.comap f).map f ≤ K := (gc_map_comap f).l_u_le _ #align ideal.map_comap_le Ideal.map_comap_le @[simp] theorem comap_top : (⊤ : Ideal S).comap f = ⊤ := (gc_map_comap f).u_top #align ideal.comap_top Ideal.comap_top @[simp] theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ := ⟨fun h => I.eq_top_iff_one.mpr (map_one f ▸ mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)), fun h => by

rw [h, comap_top]

@[simp] theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ := ⟨fun h => I.eq_top_iff_one.mpr (map_one f ▸ mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)), fun h => by

Mathlib.RingTheory.Ideal.Operations.1476_0.5qK551sG47yBciY

@[simp] theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R ι : Sort u_3 s : Set (Ideal S) ⊢ ⨅ I ∈ s, comap f I = ⨅ I ∈ comap f '' s, I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) · rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff] exact subset_span #align ideal.map_span Ideal.map_span variable {f I J K L} theorem map_le_of_le_comap : I ≤ K.comap f → I.map f ≤ K := (gc_map_comap f).l_le #align ideal.map_le_of_le_comap Ideal.map_le_of_le_comap theorem le_comap_of_map_le : I.map f ≤ K → I ≤ K.comap f := (gc_map_comap f).le_u #align ideal.le_comap_of_map_le Ideal.le_comap_of_map_le theorem le_comap_map : I ≤ (I.map f).comap f := (gc_map_comap f).le_u_l _ #align ideal.le_comap_map Ideal.le_comap_map theorem map_comap_le : (K.comap f).map f ≤ K := (gc_map_comap f).l_u_le _ #align ideal.map_comap_le Ideal.map_comap_le @[simp] theorem comap_top : (⊤ : Ideal S).comap f = ⊤ := (gc_map_comap f).u_top #align ideal.comap_top Ideal.comap_top @[simp] theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ := ⟨fun h => I.eq_top_iff_one.mpr (map_one f ▸ mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)), fun h => by rw [h, comap_top]⟩ #align ideal.comap_eq_top_iff Ideal.comap_eq_top_iff @[simp] theorem map_bot : (⊥ : Ideal R).map f = ⊥ := (gc_map_comap f).l_bot #align ideal.map_bot Ideal.map_bot variable (f I J K L) @[simp] theorem map_comap_map : ((I.map f).comap f).map f = I.map f := (gc_map_comap f).l_u_l_eq_l I #align ideal.map_comap_map Ideal.map_comap_map @[simp] theorem comap_map_comap : ((K.comap f).map f).comap f = K.comap f := (gc_map_comap f).u_l_u_eq_u K #align ideal.comap_map_comap Ideal.comap_map_comap theorem map_sup : (I ⊔ J).map f = I.map f ⊔ J.map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_sup #align ideal.map_sup Ideal.map_sup theorem comap_inf : comap f (K ⊓ L) = comap f K ⊓ comap f L := rfl #align ideal.comap_inf Ideal.comap_inf variable {ι : Sort*} theorem map_iSup (K : ι → Ideal R) : (iSup K).map f = ⨆ i, (K i).map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_iSup #align ideal.map_supr Ideal.map_iSup theorem comap_iInf (K : ι → Ideal S) : (iInf K).comap f = ⨅ i, (K i).comap f := (gc_map_comap f : GaloisConnection (map f) (comap f)).u_iInf #align ideal.comap_infi Ideal.comap_iInf theorem map_sSup (s : Set (Ideal R)) : (sSup s).map f = ⨆ I ∈ s, (I : Ideal R).map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_sSup #align ideal.map_Sup Ideal.map_sSup theorem comap_sInf (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ s, (I : Ideal S).comap f := (gc_map_comap f : GaloisConnection (map f) (comap f)).u_sInf #align ideal.comap_Inf Ideal.comap_sInf theorem comap_sInf' (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ comap f '' s, I := _root_.trans (comap_sInf f s) (by

rw [iInf_image]

theorem comap_sInf' (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ comap f '' s, I := _root_.trans (comap_sInf f s) (by

Mathlib.RingTheory.Ideal.Operations.1525_0.5qK551sG47yBciY

theorem comap_sInf' (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ comap f '' s, I

Mathlib_RingTheory_Ideal_Operations

R : Type u S : Type v F : Type u_1 inst✝¹ : Semiring R inst✝ : Semiring S rc : RingHomClass F R S f : F I J : Ideal R K L : Ideal S G : Type u_2 rcg : RingHomClass G S R ι : Sort u_3 H : IsPrime K x y : R h : x * y ∈ comap f K ⊢ f x * f y ∈ K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) · rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff] exact subset_span #align ideal.map_span Ideal.map_span variable {f I J K L} theorem map_le_of_le_comap : I ≤ K.comap f → I.map f ≤ K := (gc_map_comap f).l_le #align ideal.map_le_of_le_comap Ideal.map_le_of_le_comap theorem le_comap_of_map_le : I.map f ≤ K → I ≤ K.comap f := (gc_map_comap f).le_u #align ideal.le_comap_of_map_le Ideal.le_comap_of_map_le theorem le_comap_map : I ≤ (I.map f).comap f := (gc_map_comap f).le_u_l _ #align ideal.le_comap_map Ideal.le_comap_map theorem map_comap_le : (K.comap f).map f ≤ K := (gc_map_comap f).l_u_le _ #align ideal.map_comap_le Ideal.map_comap_le @[simp] theorem comap_top : (⊤ : Ideal S).comap f = ⊤ := (gc_map_comap f).u_top #align ideal.comap_top Ideal.comap_top @[simp] theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ := ⟨fun h => I.eq_top_iff_one.mpr (map_one f ▸ mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)), fun h => by rw [h, comap_top]⟩ #align ideal.comap_eq_top_iff Ideal.comap_eq_top_iff @[simp] theorem map_bot : (⊥ : Ideal R).map f = ⊥ := (gc_map_comap f).l_bot #align ideal.map_bot Ideal.map_bot variable (f I J K L) @[simp] theorem map_comap_map : ((I.map f).comap f).map f = I.map f := (gc_map_comap f).l_u_l_eq_l I #align ideal.map_comap_map Ideal.map_comap_map @[simp] theorem comap_map_comap : ((K.comap f).map f).comap f = K.comap f := (gc_map_comap f).u_l_u_eq_u K #align ideal.comap_map_comap Ideal.comap_map_comap theorem map_sup : (I ⊔ J).map f = I.map f ⊔ J.map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_sup #align ideal.map_sup Ideal.map_sup theorem comap_inf : comap f (K ⊓ L) = comap f K ⊓ comap f L := rfl #align ideal.comap_inf Ideal.comap_inf variable {ι : Sort*} theorem map_iSup (K : ι → Ideal R) : (iSup K).map f = ⨆ i, (K i).map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_iSup #align ideal.map_supr Ideal.map_iSup theorem comap_iInf (K : ι → Ideal S) : (iInf K).comap f = ⨅ i, (K i).comap f := (gc_map_comap f : GaloisConnection (map f) (comap f)).u_iInf #align ideal.comap_infi Ideal.comap_iInf theorem map_sSup (s : Set (Ideal R)) : (sSup s).map f = ⨆ I ∈ s, (I : Ideal R).map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_sSup #align ideal.map_Sup Ideal.map_sSup theorem comap_sInf (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ s, (I : Ideal S).comap f := (gc_map_comap f : GaloisConnection (map f) (comap f)).u_sInf #align ideal.comap_Inf Ideal.comap_sInf theorem comap_sInf' (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ comap f '' s, I := _root_.trans (comap_sInf f s) (by rw [iInf_image]) #align ideal.comap_Inf' Ideal.comap_sInf' theorem comap_isPrime [H : IsPrime K] : IsPrime (comap f K) := ⟨comap_ne_top f H.ne_top, fun {x y} h => H.mem_or_mem <| by

rwa [mem_comap, map_mul] at h

theorem comap_isPrime [H : IsPrime K] : IsPrime (comap f K) := ⟨comap_ne_top f H.ne_top, fun {x y} h => H.mem_or_mem <| by

Mathlib.RingTheory.Ideal.Operations.1529_0.5qK551sG47yBciY

theorem comap_isPrime [H : IsPrime K] : IsPrime (comap f K)

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u S✝ : Type v F : Type u_1 inst✝⁴ : Semiring R✝ inst✝³ : Semiring S✝ rc : RingHomClass F R✝ S✝ f : F I✝ J : Ideal R✝ K L : Ideal S✝ G : Type u_2 rcg : RingHomClass G S✝ R✝ ι : Sort u_3 R : Type u_4 S : Type u_5 inst✝² : CommSemiring R inst✝¹ : CommSemiring S inst✝ : Algebra R S I : Ideal R ⊢ I • ⊤ = Submodule.restrictScalars R (map (algebraMap R S) I)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton] constructor · rintro ⟨a, _, ⟨b, rfl⟩, e⟩; exact ⟨a, b, mul_comm b y ▸ e⟩ · rintro ⟨a, b, e⟩; exact ⟨a, _, ⟨b, rfl⟩, mul_comm y b ▸ e⟩ theorem isCoprime_biInf {J : ι → Ideal R} {s : Finset ι} (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by classical simp_rw [isCoprime_iff_add] at * induction s using Finset.induction with | empty => simp | @insert i s _ hs => rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top] set K := ⨅ j ∈ s, J j calc 1 = I + K := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm _ = I + K*(I + J i) := by rw [hf i (Finset.mem_insert_self i s), mul_one] _ = (1+K)*I + K*J i := by ring _ ≤ I + K ⊓ J i := add_le_add mul_le_left mul_le_inf /-- The radical of an ideal `I` consists of the elements `r` such that `r ^ n ∈ I` for some `n`. -/ def radical (I : Ideal R) : Ideal R where carrier := { r | ∃ n : ℕ, r ^ n ∈ I } zero_mem' := ⟨1, (pow_one (0 : R)).symm ▸ I.zero_mem⟩ add_mem' := fun {x y} ⟨m, hxmi⟩ ⟨n, hyni⟩ => ⟨m + n, (add_pow x y (m + n)).symm ▸ I.sum_mem <| show ∀ c ∈ Finset.range (Nat.succ (m + n)), x ^ c * y ^ (m + n - c) * Nat.choose (m + n) c ∈ I from fun c _ => Or.casesOn (le_total c m) (fun hcm => I.mul_mem_right _ <| I.mul_mem_left _ <| Nat.add_comm n m ▸ (add_tsub_assoc_of_le hcm n).symm ▸ (pow_add y n (m - c)).symm ▸ I.mul_mem_right _ hyni) (fun hmc => I.mul_mem_right _ <| I.mul_mem_right _ <| add_tsub_cancel_of_le hmc ▸ (pow_add x m (c - m)).symm ▸ I.mul_mem_right _ hxmi)⟩ -- Porting note: Below gives weird errors without `by exact` smul_mem' {r s} := by exact fun ⟨n, h⟩ ↦ ⟨n, (mul_pow r s n).symm ▸ I.mul_mem_left (r ^ n) h⟩ #align ideal.radical Ideal.radical /-- An ideal is radical if it contains its radical. -/ def IsRadical (I : Ideal R) : Prop := I.radical ≤ I #align ideal.is_radical Ideal.IsRadical theorem le_radical : I ≤ radical I := fun r hri => ⟨1, (pow_one r).symm ▸ hri⟩ #align ideal.le_radical Ideal.le_radical /-- An ideal is radical iff it is equal to its radical. -/ theorem radical_eq_iff : I.radical = I ↔ I.IsRadical := by rw [le_antisymm_iff, and_iff_left le_radical, IsRadical] #align ideal.radical_eq_iff Ideal.radical_eq_iff alias ⟨_, IsRadical.radical⟩ := radical_eq_iff #align ideal.is_radical.radical Ideal.IsRadical.radical variable (R) theorem radical_top : (radical ⊤ : Ideal R) = ⊤ := (eq_top_iff_one _).2 ⟨0, Submodule.mem_top⟩ #align ideal.radical_top Ideal.radical_top variable {R} theorem radical_mono (H : I ≤ J) : radical I ≤ radical J := fun _ ⟨n, hrni⟩ => ⟨n, H hrni⟩ #align ideal.radical_mono Ideal.radical_mono variable (I) theorem radical_isRadical : (radical I).IsRadical := fun r ⟨n, k, hrnki⟩ => ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ #align ideal.radical_is_radical Ideal.radical_isRadical @[simp] theorem radical_idem : radical (radical I) = radical I := (radical_isRadical I).radical #align ideal.radical_idem Ideal.radical_idem variable {I} theorem IsRadical.radical_le_iff (hJ : J.IsRadical) : I.radical ≤ J ↔ I ≤ J := ⟨le_trans le_radical, fun h => hJ.radical ▸ radical_mono h⟩ #align ideal.is_radical.radical_le_iff Ideal.IsRadical.radical_le_iff theorem radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J := (radical_isRadical J).radical_le_iff #align ideal.radical_le_radical_iff Ideal.radical_le_radical_iff theorem radical_eq_top : radical I = ⊤ ↔ I = ⊤ := ⟨fun h => (eq_top_iff_one _).2 <| let ⟨n, hn⟩ := (eq_top_iff_one _).1 h @one_pow R _ n ▸ hn, fun h => h.symm ▸ radical_top R⟩ #align ideal.radical_eq_top Ideal.radical_eq_top theorem IsPrime.isRadical (H : IsPrime I) : I.IsRadical := fun _ ⟨n, hrni⟩ => H.mem_of_pow_mem n hrni #align ideal.is_prime.is_radical Ideal.IsPrime.isRadical theorem IsPrime.radical (H : IsPrime I) : radical I = I := IsRadical.radical H.isRadical #align ideal.is_prime.radical Ideal.IsPrime.radical variable (I J) theorem radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) := le_antisymm (radical_mono <| sup_le_sup le_radical le_radical) <| radical_le_radical_iff.2 <| sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) #align ideal.radical_sup Ideal.radical_sup theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J := le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right)) fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm, (pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩ #align ideal.radical_inf Ideal.radical_inf theorem radical_mul : radical (I * J) = radical I ⊓ radical J := by refine le_antisymm ?_ fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ => ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩ have := radical_mono <| @mul_le_inf _ _ I J simp_rw [radical_inf I J] at this assumption #align ideal.radical_mul Ideal.radical_mul variable {I J} theorem IsPrime.radical_le_iff (hJ : IsPrime J) : I.radical ≤ J ↔ I ≤ J := IsRadical.radical_le_iff hJ.isRadical #align ideal.is_prime.radical_le_iff Ideal.IsPrime.radical_le_iff theorem radical_eq_sInf (I : Ideal R) : radical I = sInf { J : Ideal R | I ≤ J ∧ IsPrime J } := le_antisymm (le_sInf fun J hJ ↦ hJ.2.radical_le_iff.2 hJ.1) fun r hr ↦ by_contradiction fun hri ↦ let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partialOrder₀ { K : Ideal R | r ∉ radical K } (fun c hc hcc y hyc => ⟨sSup c, fun ⟨n, hrnc⟩ => let ⟨y, hyc, hrny⟩ := (Submodule.mem_sSup_of_directed ⟨y, hyc⟩ hcc.directedOn).1 hrnc hc hyc ⟨n, hrny⟩, fun z => le_sSup⟩) I hri have : ∀ (x) (_ : x ∉ m), r ∈ radical (m ⊔ span {x}) := fun x hxm => by_contradiction fun hrmx => hxm <| hm (m ⊔ span {x}) hrmx le_sup_left ▸ (le_sup_right : _ ≤ m ⊔ span {x}) (subset_span <| Set.mem_singleton _) have : IsPrime m := ⟨by rintro rfl; rw [radical_top] at hrm; exact hrm trivial, fun {x y} hxym => or_iff_not_imp_left.2 fun hxm => by_contradiction fun hym => let ⟨n, hrn⟩ := this _ hxm let ⟨p, hpm, q, hq, hpqrn⟩ := Submodule.mem_sup.1 hrn let ⟨c, hcxq⟩ := mem_span_singleton'.1 hq let ⟨k, hrk⟩ := this _ hym let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk let ⟨d, hdyg⟩ := mem_span_singleton'.1 hg hrm ⟨n + k, by rw [pow_add, ← hpqrn, ← hcxq, ← hfgrk, ← hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, ← mul_assoc]; refine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))⟩⟩ hrm <| this.radical.symm ▸ (sInf_le ⟨him, this⟩ : sInf { J : Ideal R | I ≤ J ∧ IsPrime J } ≤ m) hr #align ideal.radical_eq_Inf Ideal.radical_eq_sInf theorem isRadical_bot_of_noZeroDivisors {R} [CommSemiring R] [NoZeroDivisors R] : (⊥ : Ideal R).IsRadical := fun _ hx => hx.recOn fun _ hn => pow_eq_zero hn #align ideal.is_radical_bot_of_no_zero_divisors Ideal.isRadical_bot_of_noZeroDivisors @[simp] theorem radical_bot_of_noZeroDivisors {R : Type u} [CommSemiring R] [NoZeroDivisors R] : radical (⊥ : Ideal R) = ⊥ := eq_bot_iff.2 isRadical_bot_of_noZeroDivisors #align ideal.radical_bot_of_no_zero_divisors Ideal.radical_bot_of_noZeroDivisors instance : IdemCommSemiring (Ideal R) := inferInstance variable (R) theorem top_pow (n : ℕ) : (⊤ ^ n : Ideal R) = ⊤ := Nat.recOn n one_eq_top fun n ih => by rw [pow_succ, ih, top_mul] #align ideal.top_pow Ideal.top_pow variable {R} variable (I) theorem radical_pow (n : ℕ) (H : n > 0) : radical (I ^ n) = radical I := Nat.recOn n (Not.elim (by decide)) (fun n ih H => Or.casesOn (lt_or_eq_of_le <| Nat.le_of_lt_succ H) (fun H => calc radical (I ^ (n + 1)) = radical I ⊓ radical (I ^ n) := by rw [pow_succ] exact radical_mul _ _ _ = radical I ⊓ radical I := by rw [ih H] _ = radical I := inf_idem ) fun H => H ▸ (pow_one I).symm ▸ rfl) H #align ideal.radical_pow Ideal.radical_pow theorem IsPrime.mul_le {I J P : Ideal R} (hp : IsPrime P) : I * J ≤ P ↔ I ≤ P ∨ J ≤ P := by rw [or_comm, Ideal.mul_le] simp_rw [hp.mul_mem_iff_mem_or_mem, SetLike.le_def, ← forall_or_left, or_comm, forall_or_left] #align ideal.is_prime.mul_le Ideal.IsPrime.mul_le theorem IsPrime.inf_le {I J P : Ideal R} (hp : IsPrime P) : I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P := ⟨fun h ↦ hp.mul_le.1 <| mul_le_inf.trans h, fun h ↦ h.elim inf_le_left.trans inf_le_right.trans⟩ #align ideal.is_prime.inf_le Ideal.IsPrime.inf_le theorem IsPrime.multiset_prod_le {s : Multiset (Ideal R)} {P : Ideal R} (hp : IsPrime P) : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P := s.induction_on (by simp [hp.ne_top]) fun I s ih ↦ by simp [hp.mul_le, ih] #align ideal.is_prime.multiset_prod_le Ideal.IsPrime.multiset_prod_le theorem IsPrime.multiset_prod_map_le {s : Multiset ι} (f : ι → Ideal R) {P : Ideal R} (hp : IsPrime P) : (s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P := by simp_rw [hp.multiset_prod_le, Multiset.mem_map, exists_exists_and_eq_and] #align ideal.is_prime.multiset_prod_map_le Ideal.IsPrime.multiset_prod_map_le theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P := hp.multiset_prod_map_le f #align ideal.is_prime.prod_le Ideal.IsPrime.prod_le theorem IsPrime.inf_le' {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) : s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P := ⟨fun h ↦ hp.prod_le.1 <| prod_le_inf.trans h, fun ⟨_, his, hip⟩ ↦ (Finset.inf_le his).trans hip⟩ #align ideal.is_prime.inf_le' Ideal.IsPrime.inf_le' -- Porting note: needed to add explicit coercions (· : Set R). theorem subset_union {R : Type u} [Ring R] {I J K : Ideal R} : (I : Set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K := AddSubgroupClass.subset_union #align ideal.subset_union Ideal.subset_union theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι} (hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i := by suffices ((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i from ⟨this, fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h => Or.casesOn h (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _)) fun ⟨i, his, hi⟩ => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _; exact Set.subset_biUnion_of_mem (u := fun x ↦ (f x : Set R)) (Finset.mem_coe.2 his)⟩ generalize hn : s.card = n; intro h induction' n with n ih generalizing a b s · clear hp rw [Finset.card_eq_zero] at hn subst hn rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff] classical replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n := Finset.card_eq_succ.1 hn rcases hn with ⟨i, t, hit, rfl, hn⟩ replace hp : IsPrime (f i) ∧ ∀ x ∈ t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp by_cases Ht : ∃ j ∈ t, f j ≤ f i · obtain ⟨j, hjt, hfji⟩ : ∃ j ∈ t, f j ≤ f i := Ht obtain ⟨u, hju, rfl⟩ : ∃ u, j ∉ u ∧ insert j u = t := ⟨t.erase j, t.not_mem_erase j, Finset.insert_erase hjt⟩ have hp' : ∀ k ∈ insert i u, IsPrime (f k) := by rw [Finset.forall_mem_insert] at hp ⊢ exact ⟨hp.1, hp.2.2⟩ have hiu : i ∉ u := mt Finset.mem_insert_of_mem hit have hn' : (insert i u).card = n := by rwa [Finset.card_insert_of_not_mem] at hn ⊢ exacts [hiu, hju] have h' : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ k ∈ (↑(insert i u) : Set ι), f k := by rw [Finset.coe_insert] at h ⊢ rw [Finset.coe_insert] at h simp only [Set.biUnion_insert] at h ⊢ rw [← Set.union_assoc (f i : Set R)] at h erw [Set.union_eq_self_of_subset_right hfji] at h exact h specialize ih hp' hn' h' refine' ih.imp id (Or.imp id (Exists.imp fun k => _)) exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id by_cases Ha : f a ≤ f i · have h' : (I : Set R) ⊆ f i ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_right_comm (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Ha] at h exact h specialize ih hp.2 hn h' right rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩ · exact Or.inl ih · exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ by_cases Hb : f b ≤ f i · have h' : (I : Set R) ⊆ f a ∪ f i ∪ ⋃ j ∈ (↑t : Set ι), f j := by rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_assoc, Set.union_assoc (f a : Set R)] at h erw [Set.union_eq_self_of_subset_left Hb] at h exact h specialize ih hp.2 hn h' rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · exact Or.inl ih · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, ih⟩) · exact Or.inr (Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩) by_cases Hi : I ≤ f i · exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩) have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by simp only [hp.1.inf_le, hp.1.inf_le', not_or] exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩ rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩ by_cases HI : (I : Set R) ⊆ f a ∪ f b ∪ ⋃ j ∈ (↑t : Set ι), f j · specialize ih hp.2 hn HI rcases ih with (ih | ih | ⟨k, hkt, ih⟩) · left exact ih · right left exact ih · right right exact ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ exfalso rcases Set.not_subset.1 HI with ⟨s, hsI, hs⟩ rw [Finset.coe_insert, Set.biUnion_insert] at h have hsi : s ∈ f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs) rcases h (I.add_mem hrI hsI) with (⟨ha | hb⟩ | hi | ht) · exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s ▸ (f a).sub_mem ha hra) · exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s ▸ (f b).sub_mem hb hrb) · exact hri (add_sub_cancel r s ▸ (f i).sub_mem hi hsi) · rw [Set.mem_iUnion₂] at ht rcases ht with ⟨j, hjt, hj⟩ simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s ▸ (f j).sub_mem hj <| hr j hjt) #align ideal.subset_union_prime' Ideal.subset_union_prime' /-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/ theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι) (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} : ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i := suffices ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i by have aux := fun h => (bex_def.2 <| this h) simp_rw [exists_prop] at aux refine ⟨aux, fun ⟨i, his, hi⟩ ↦ Set.Subset.trans hi ?_⟩ apply Set.subset_biUnion_of_mem (show i ∈ (↑s : Set ι) from his) fun h : (I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i => by classical by_cases has : a ∈ s · obtain ⟨t, hat, rfl⟩ : ∃ t, a ∉ t ∧ insert a t = s := ⟨s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has⟩ by_cases hbt : b ∈ t · obtain ⟨u, hbu, rfl⟩ : ∃ u, b ∉ u ∧ insert b u = t := ⟨t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt⟩ have hp' : ∀ i ∈ u, IsPrime (f i) := by intro i hiu refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, ← Set.union_assoc, subset_union_prime' hp'] at h rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] · have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f a : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] · by_cases hbs : b ∈ s · obtain ⟨t, hbt, rfl⟩ : ∃ t, b ∉ t ∧ insert b t = s := ⟨s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f b : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] rcases s.eq_empty_or_nonempty with hse | hsne · subst hse rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h have : (I : Set R) ≠ ∅ := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) exact absurd h this · cases' hsne.bex with i his obtain ⟨t, _, rfl⟩ : ∃ t, i ∉ t ∧ insert i t = s := ⟨s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his⟩ have hp' : ∀ j ∈ t, IsPrime (f j) := by intro j hj refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *] rw [Finset.coe_insert, Set.biUnion_insert, ← Set.union_self (f i : Set R), subset_union_prime' hp', ← or_assoc, or_self_iff] at h rwa [Finset.exists_mem_insert] #align ideal.subset_union_prime Ideal.subset_union_prime section Dvd /-- If `I` divides `J`, then `I` contains `J`. In a Dedekind domain, to divide and contain are equivalent, see `Ideal.dvd_iff_le`. -/ theorem le_of_dvd {I J : Ideal R} : I ∣ J → J ≤ I | ⟨_, h⟩ => h.symm ▸ le_trans mul_le_inf inf_le_left #align ideal.le_of_dvd Ideal.le_of_dvd theorem isUnit_iff {I : Ideal R} : IsUnit I ↔ I = ⊤ := isUnit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸ ⟨fun h => eq_top_iff.mpr (Ideal.le_of_dvd h), fun h => ⟨⊤, by rw [mul_top, h]⟩⟩) #align ideal.is_unit_iff Ideal.isUnit_iff instance uniqueUnits : Unique (Ideal R)ˣ where default := 1 uniq u := Units.ext (show (u : Ideal R) = 1 by rw [isUnit_iff.mp u.isUnit, one_eq_top]) #align ideal.unique_units Ideal.uniqueUnits end Dvd end MulAndRadical section MapAndComap variable {R : Type u} {S : Type v} section Semiring variable {F : Type*} [Semiring R] [Semiring S] variable [rc : RingHomClass F R S] variable (f : F) variable {I J : Ideal R} {K L : Ideal S} /-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than the image itself. -/ def map (I : Ideal R) : Ideal S := span (f '' I) #align ideal.map Ideal.map /-- `I.comap f` is the preimage of `I` under `f`. -/ def comap (I : Ideal S) : Ideal R where carrier := f ⁻¹' I add_mem' {x y} hx hy := by simp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy ⊢ exact add_mem hx hy zero_mem' := by simp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem] smul_mem' c x hx := by simp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at * exact mul_mem_left I _ hx #align ideal.comap Ideal.comap -- Porting note: new theorem -- @[simp] -- Porting note: TODO enable simp after the port theorem coe_comap (I : Ideal S) : (comap f I : Set R) = f ⁻¹' I := rfl variable {f} theorem map_mono (h : I ≤ J) : map f I ≤ map f J := span_mono <| Set.image_subset _ h #align ideal.map_mono Ideal.map_mono theorem mem_map_of_mem (f : F) {I : Ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I := subset_span ⟨x, h, rfl⟩ #align ideal.mem_map_of_mem Ideal.mem_map_of_mem theorem apply_coe_mem_map (f : F) (I : Ideal R) (x : I) : f x ∈ I.map f := mem_map_of_mem f x.2 #align ideal.apply_coe_mem_map Ideal.apply_coe_mem_map theorem map_le_iff_le_comap : map f I ≤ K ↔ I ≤ comap f K := span_le.trans Set.image_subset_iff #align ideal.map_le_iff_le_comap Ideal.map_le_iff_le_comap @[simp] theorem mem_comap {x} : x ∈ comap f K ↔ f x ∈ K := Iff.rfl #align ideal.mem_comap Ideal.mem_comap theorem comap_mono (h : K ≤ L) : comap f K ≤ comap f L := Set.preimage_mono fun _ hx => h hx #align ideal.comap_mono Ideal.comap_mono variable (f) theorem comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ := (ne_top_iff_one _).2 <| by rw [mem_comap, map_one]; exact (ne_top_iff_one _).1 hK #align ideal.comap_ne_top Ideal.comap_ne_top variable {G : Type*} [rcg : RingHomClass G S R] theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) : I.map f ≤ I.comap g := by refine' Ideal.span_le.2 _ rintro x ⟨x, hx, rfl⟩ rw [SetLike.mem_coe, mem_comap, hf hx] exact hx #align ideal.map_le_comap_of_inv_on Ideal.map_le_comap_of_inv_on theorem comap_le_map_of_inv_on (g : G) (I : Ideal S) (hf : Set.LeftInvOn g f (f ⁻¹' I)) : I.comap f ≤ I.map g := fun x (hx : f x ∈ I) => hf hx ▸ Ideal.mem_map_of_mem g hx #align ideal.comap_le_map_of_inv_on Ideal.comap_le_map_of_inv_on /-- The `Ideal` version of `Set.image_subset_preimage_of_inverse`. -/ theorem map_le_comap_of_inverse (g : G) (I : Ideal R) (h : Function.LeftInverse g f) : I.map f ≤ I.comap g := map_le_comap_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.map_le_comap_of_inverse Ideal.map_le_comap_of_inverse /-- The `Ideal` version of `Set.preimage_subset_image_of_inverse`. -/ theorem comap_le_map_of_inverse (g : G) (I : Ideal S) (h : Function.LeftInverse g f) : I.comap f ≤ I.map g := comap_le_map_of_inv_on _ _ _ <| h.leftInvOn _ #align ideal.comap_le_map_of_inverse Ideal.comap_le_map_of_inverse instance IsPrime.comap [hK : K.IsPrime] : (comap f K).IsPrime := ⟨comap_ne_top _ hK.1, fun {x y} => by simp only [mem_comap, map_mul]; apply hK.2⟩ #align ideal.is_prime.comap Ideal.IsPrime.comap variable (I J K L) theorem map_top : map f ⊤ = ⊤ := (eq_top_iff_one _).2 <| subset_span ⟨1, trivial, map_one f⟩ #align ideal.map_top Ideal.map_top theorem gc_map_comap : GaloisConnection (Ideal.map f) (Ideal.comap f) := fun _ _ => Ideal.map_le_iff_le_comap #align ideal.gc_map_comap Ideal.gc_map_comap @[simp] theorem comap_id : I.comap (RingHom.id R) = I := Ideal.ext fun _ => Iff.rfl #align ideal.comap_id Ideal.comap_id @[simp] theorem map_id : I.map (RingHom.id R) = I := (gc_map_comap (RingHom.id R)).l_unique GaloisConnection.id comap_id #align ideal.map_id Ideal.map_id theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl #align ideal.comap_comap Ideal.comap_comap theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) : (I.map f).map g = I.map (g.comp f) := ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ => comap_comap _ _ #align ideal.map_map Ideal.map_map theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by refine (Submodule.span_eq_of_le _ ?_ ?_).symm · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx) · rw [map_le_iff_le_comap, span_le, coe_comap, ← Set.image_subset_iff] exact subset_span #align ideal.map_span Ideal.map_span variable {f I J K L} theorem map_le_of_le_comap : I ≤ K.comap f → I.map f ≤ K := (gc_map_comap f).l_le #align ideal.map_le_of_le_comap Ideal.map_le_of_le_comap theorem le_comap_of_map_le : I.map f ≤ K → I ≤ K.comap f := (gc_map_comap f).le_u #align ideal.le_comap_of_map_le Ideal.le_comap_of_map_le theorem le_comap_map : I ≤ (I.map f).comap f := (gc_map_comap f).le_u_l _ #align ideal.le_comap_map Ideal.le_comap_map theorem map_comap_le : (K.comap f).map f ≤ K := (gc_map_comap f).l_u_le _ #align ideal.map_comap_le Ideal.map_comap_le @[simp] theorem comap_top : (⊤ : Ideal S).comap f = ⊤ := (gc_map_comap f).u_top #align ideal.comap_top Ideal.comap_top @[simp] theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ := ⟨fun h => I.eq_top_iff_one.mpr (map_one f ▸ mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)), fun h => by rw [h, comap_top]⟩ #align ideal.comap_eq_top_iff Ideal.comap_eq_top_iff @[simp] theorem map_bot : (⊥ : Ideal R).map f = ⊥ := (gc_map_comap f).l_bot #align ideal.map_bot Ideal.map_bot variable (f I J K L) @[simp] theorem map_comap_map : ((I.map f).comap f).map f = I.map f := (gc_map_comap f).l_u_l_eq_l I #align ideal.map_comap_map Ideal.map_comap_map @[simp] theorem comap_map_comap : ((K.comap f).map f).comap f = K.comap f := (gc_map_comap f).u_l_u_eq_u K #align ideal.comap_map_comap Ideal.comap_map_comap theorem map_sup : (I ⊔ J).map f = I.map f ⊔ J.map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_sup #align ideal.map_sup Ideal.map_sup theorem comap_inf : comap f (K ⊓ L) = comap f K ⊓ comap f L := rfl #align ideal.comap_inf Ideal.comap_inf variable {ι : Sort*} theorem map_iSup (K : ι → Ideal R) : (iSup K).map f = ⨆ i, (K i).map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_iSup #align ideal.map_supr Ideal.map_iSup theorem comap_iInf (K : ι → Ideal S) : (iInf K).comap f = ⨅ i, (K i).comap f := (gc_map_comap f : GaloisConnection (map f) (comap f)).u_iInf #align ideal.comap_infi Ideal.comap_iInf theorem map_sSup (s : Set (Ideal R)) : (sSup s).map f = ⨆ I ∈ s, (I : Ideal R).map f := (gc_map_comap f : GaloisConnection (map f) (comap f)).l_sSup #align ideal.map_Sup Ideal.map_sSup theorem comap_sInf (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ s, (I : Ideal S).comap f := (gc_map_comap f : GaloisConnection (map f) (comap f)).u_sInf #align ideal.comap_Inf Ideal.comap_sInf theorem comap_sInf' (s : Set (Ideal S)) : (sInf s).comap f = ⨅ I ∈ comap f '' s, I := _root_.trans (comap_sInf f s) (by rw [iInf_image]) #align ideal.comap_Inf' Ideal.comap_sInf' theorem comap_isPrime [H : IsPrime K] : IsPrime (comap f K) := ⟨comap_ne_top f H.ne_top, fun {x y} h => H.mem_or_mem <| by rwa [mem_comap, map_mul] at h⟩ #align ideal.comap_is_prime Ideal.comap_isPrime variable {I J K L} theorem map_inf_le : map f (I ⊓ J) ≤ map f I ⊓ map f J := (gc_map_comap f : GaloisConnection (map f) (comap f)).monotone_l.map_inf_le _ _ #align ideal.map_inf_le Ideal.map_inf_le theorem le_comap_sup : comap f K ⊔ comap f L ≤ comap f (K ⊔ L) := (gc_map_comap f : GaloisConnection (map f) (comap f)).monotone_u.le_map_sup _ _ #align ideal.le_comap_sup Ideal.le_comap_sup @[simp] theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S] (I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by

refine' le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx => Submodule.span_induction hx _ _ _ _

@[simp] theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S] (I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R := by

Mathlib.RingTheory.Ideal.Operations.1543_0.5qK551sG47yBciY

@[simp] theorem smul_top_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] [Algebra R S] (I : Ideal R) : I • (⊤ : Submodule R S) = (I.map (algebraMap R S)).restrictScalars R

Mathlib_RingTheory_Ideal_Operations