Loogle!

Result

Found 304 declarations mentioning Ideal.comap. Of these, only the first 200 are shown.

Ideal.comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (I : Ideal S) : Ideal R
Ideal.comap_isPrime 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (K : Ideal S) [RingHomClass F R S] [H : K.IsPrime] : (Ideal.comap f K).IsPrime
Ideal.instIsTwoSidedComap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {K : Ideal S} [RingHomClass F R S] [K.IsTwoSided] : (Ideal.comap f K).IsTwoSided
Ideal.IsPrime.comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {K : Ideal S} [RingHomClass F R S] [hK : K.IsPrime] : (Ideal.comap f K).IsPrime
Ideal.comap_id 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} [Semiring R] (I : Ideal R) : Ideal.comap (RingHom.id R) I = I
Ideal.comap_injective_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) : Function.Injective (Ideal.comap f)
Ideal.IsMaximal.comap_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {K : Ideal S} : K.IsMaximal → (Ideal.comap f K).IsMaximal
Ideal.IsRadical.comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [CommSemiring R] [CommSemiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (hK : K.IsRadical) : (Ideal.comap f K).IsRadical
Ideal.isMaximal_comap_iff_of_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {K : Ideal S} : (Ideal.comap f K).IsMaximal ↔ K.IsMaximal
Ideal.comap_map_of_bijective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {I : Ideal R} : Ideal.comap f (Ideal.map f I) = I
Ideal.map_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) (I : Ideal S) : Ideal.map f (Ideal.comap f I) = I
RingHom.ker_eq_comap_bot 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] (f : F) : RingHom.ker f = Ideal.comap f ⊥
Ideal.comap_top 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} [RingHomClass F R S] : Ideal.comap f ⊤ = ⊤
Ideal.map_comap_map 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (I : Ideal R) [RingHomClass F R S] : Ideal.map f (Ideal.comap f (Ideal.map f I)) = Ideal.map f I
Ideal.comap_isMaximal_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {K : Ideal S} [H : K.IsMaximal] : (Ideal.comap f K).IsMaximal
Ideal.comap_map_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (K : Ideal S) [RingHomClass F R S] : Ideal.comap f (Ideal.map f (Ideal.comap f K)) = Ideal.comap f K
Ideal.ker_le_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {S : Type u_2} {F : Type u_3} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] {K : Ideal S} (f : F) : RingHom.ker f ≤ Ideal.comap f K
Ideal.le_comap_map 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {I : Ideal R} [RingHomClass F R S] : I ≤ Ideal.comap f (Ideal.map f I)
Ideal.map_comap_le 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {K : Ideal S} [RingHomClass F R S] : Ideal.map f (Ideal.comap f K) ≤ K
Ideal.comap_isMaximal_of_equiv 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {E : Type u_4} [EquivLike E R S] [RingEquivClass E R S] (e : E) {p : Ideal S} [hp : p.IsMaximal] : (Ideal.comap e p).IsMaximal
Ideal.comap_radical 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [CommSemiring R] [CommSemiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (K : Ideal S) : Ideal.comap f K.radical = (Ideal.comap f K).radical
Ideal.comap_ne_top 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {K : Ideal S} [RingHomClass F R S] (hK : K ≠ ⊤) : Ideal.comap f K ≠ ⊤
Ideal.comap_bot_of_injective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Injective ⇑f) : Ideal.comap f ⊥ = ⊥
Ideal.comap_eq_top_iff 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} [RingHomClass F R S] {I : Ideal S} : Ideal.comap f I = ⊤ ↔ I = ⊤
Ideal.gc_map_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] : GaloisConnection (Ideal.map f) (Ideal.comap f)
Ideal.coe_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (I : Ideal S) : ↑(Ideal.comap f I) = ⇑f ⁻¹' ↑I
Ideal.comap_bot_le_of_injective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {I : Ideal R} [RingHomClass F R S] (hf : Function.Injective ⇑f) : Ideal.comap f ⊥ ≤ I
Ideal.map_comap_eq_self_of_equiv 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {E : Type u_4} [EquivLike E R S] [RingEquivClass E R S] (e : E) (I : Ideal S) : Ideal.map e (Ideal.comap e I) = I
Ideal.comap_iInf 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] {ι : Sort u_3} (K : ι → Ideal S) : Ideal.comap f (iInf K) = ⨅ i, Ideal.comap f (K i)
Ideal.comap_idₐ 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_3} {S : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] (I : Ideal S) : Ideal.comap (AlgHom.id R S) I = I
Ideal.giMapComap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) : GaloisInsertion (Ideal.map f) (Ideal.comap f)
Ideal.comap_coe 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (I : Ideal S) : Ideal.comap (↑f) I = Ideal.comap f I
Ideal.mem_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {K : Ideal S} [RingHomClass F R S] {x : R} : x ∈ Ideal.comap f K ↔ f x ∈ K
Ideal.comap_le_map_of_inverse 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {G : Type u_2} [FunLike G S R] [RingHomClass F R S] (g : G) (I : Ideal S) (h : Function.LeftInverse ⇑g ⇑f) : Ideal.comap f I ≤ Ideal.map g I
Ideal.map_le_comap_of_inverse 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {G : Type u_2} [FunLike G S R] [RingHomClass G S R] (g : G) (I : Ideal R) (h : Function.LeftInverse ⇑g ⇑f) : Ideal.map f I ≤ Ideal.comap g I
Ideal.comap_inf 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (K L : Ideal S) [RingHomClass F R S] : Ideal.comap f (K ⊓ L) = Ideal.comap f K ⊓ Ideal.comap f L
Ideal.map_iInf_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] {ι : Sort u_3} (hf : Function.Surjective ⇑f) (K : ι → Ideal S) : Ideal.map f (⨅ i, Ideal.comap f (K i)) = iInf K
Ideal.le_comap_of_map_le 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {I : Ideal R} {K : Ideal S} [RingHomClass F R S] : Ideal.map f I ≤ K → I ≤ Ideal.comap f K
Ideal.map_le_of_le_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {I : Ideal R} {K : Ideal S} [RingHomClass F R S] : I ≤ Ideal.comap f K → Ideal.map f I ≤ K
Ideal.comap_mono 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {K L : Ideal S} [RingHomClass F R S] (h : K ≤ L) : Ideal.comap f K ≤ Ideal.comap f L
Ideal.map_le_iff_le_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {I : Ideal R} {K : Ideal S} [RingHomClass F R S] : Ideal.map f I ≤ K ↔ I ≤ Ideal.comap f K
Ideal.map_inf_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) (I J : Ideal S) : Ideal.map f (Ideal.comap f I ⊓ Ideal.comap f J) = I ⊓ J
Ideal.le_map_of_comap_le_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {I : Ideal R} {K : Ideal S} [RingHomClass F R S] (hf : Function.Surjective ⇑f) : Ideal.comap f K ≤ I → K ≤ Ideal.map f I
Ideal.comap_le_iff_le_map 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Bijective ⇑f) {I : Ideal R} {K : Ideal S} : Ideal.comap f K ≤ I ↔ K ≤ Ideal.map f I
Ideal.comap_le_comap_iff_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) (I J : Ideal S) : Ideal.comap f I ≤ Ideal.comap f J ↔ I ≤ J
Ideal.map_le_comap_of_inv_on 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {G : Type u_2} [FunLike G S R] [RingHomClass G S R] (g : G) (I : Ideal R) (hf : Set.LeftInvOn ⇑g ⇑f ↑I) : Ideal.map f I ≤ Ideal.comap g I
Ideal.IsMaximal.comap_piEvalRingHom 📋 Mathlib.RingTheory.Ideal.Maps
{ι : Type u_4} {R : ι → Type u_5} [(i : ι) → Semiring (R i)] {i : ι} {I : Ideal (R i)} (h : I.IsMaximal) : (Ideal.comap (Pi.evalRingHom R i) I).IsMaximal
Ideal.coheight_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal S) : Order.coheight (Ideal.comap f I) = Order.coheight I
Ideal.eq_bot_of_comap_eq_bot' 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) {I : Ideal S} (h : Ideal.comap f I = ⊥) : I = ⊥
Ideal.comap_le_map_of_inv_on 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {G : Type u_2} [FunLike G S R] [RingHomClass F R S] (g : G) (I : Ideal S) (hf : Set.LeftInvOn (⇑g) (⇑f) (⇑f ⁻¹' ↑I)) : Ideal.comap f I ≤ Ideal.map g I
Ideal.map_primeCompl_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) (p : Ideal S) [p.IsPrime] : Submonoid.map f (Ideal.comap f p).primeCompl = p.primeCompl
Ideal.map_iSup_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] {ι : Sort u_3} (hf : Function.Surjective ⇑f) (K : ι → Ideal S) : Ideal.map f (⨆ i, Ideal.comap f (K i)) = iSup K
Ideal.comap_sInf 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (s : Set (Ideal S)) : Ideal.comap f (sInf s) = ⨅ I ∈ s, Ideal.comap f I
Ideal.comap_finsetInf 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] {ι : Type u_4} (s : Finset ι) (K : ι → Ideal S) : Ideal.comap f (s.inf K) = s.inf fun i => Ideal.comap f (K i)
Ideal.map_sup_comap_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) (I J : Ideal S) : Ideal.map f (Ideal.comap f I ⊔ Ideal.comap f J) = I ⊔ J
Ideal.comap_map_of_surjective' 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {S : Type u_2} {F : Type u_3} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal R) : Ideal.comap f (Ideal.map f I) = I ⊔ RingHom.ker f
Ideal.le_comap_sup 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {K L : Ideal S} [RingHomClass F R S] : Ideal.comap f K ⊔ Ideal.comap f L ≤ Ideal.comap f (K ⊔ L)
RingHom.comap_ker 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S →+* R) (g : T →+* S) : Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)
Ideal.comap_comap 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_3} [Semiring T] {I : Ideal T} (f : R →+* S) (g : S →+* T) : Ideal.comap f (Ideal.comap g I) = Ideal.comap (g.comp f) I
Ideal.comap_sInf' 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (s : Set (Ideal S)) : Ideal.comap f (sInf s) = ⨅ I ∈ Ideal.comap f '' s, I
Ideal.comap_map_of_surjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal R) : Ideal.comap f (Ideal.map f I) = I ⊔ Ideal.comap f ⊥
Ideal.exists_ideal_comap_le_prime 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {F : Type u_1} [Semiring R] {S : Type u_2} [CommSemiring S] [FunLike F R S] [RingHomClass F R S] {f : F} (P : Ideal R) [P.IsPrime] (I : Ideal S) (le : Ideal.comap f I ≤ P) : ∃ Q ≥ I, Q.IsPrime ∧ Ideal.comap f Q ≤ P
Module.comap_annihilator 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {R₀ : Type u_4} [CommSemiring R₀] [Module R₀ M] [Algebra R₀ R] [IsScalarTower R₀ R M] : Ideal.comap (algebraMap R₀ R) (Module.annihilator R M) = Module.annihilator R₀ M
Ideal.le_comap_pow 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [CommSemiring R] [CommSemiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (n : ℕ) : Ideal.comap f K ^ n ≤ Ideal.comap f (K ^ n)
Ideal.comap_map_eq_self_of_isMaximal 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) {p : Ideal R} [hP' : p.IsMaximal] (hP : Ideal.map f p ≠ ⊤) : Ideal.comap f (Ideal.map f p) = p
Ideal.comap_map_eq_self_iff_of_isPrime 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} [CommSemiring R] {S : Type u_2} [CommSemiring S] {f : R →+* S} (p : Ideal R) [p.IsPrime] : Ideal.comap f (Ideal.map f p) = p ↔ ∃ q, q.IsPrime ∧ Ideal.comap f q = p
Ideal.orderEmbeddingOfSurjective_apply 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] (hf : Function.Surjective ⇑f) (I : Ideal S) : (Ideal.orderEmbeddingOfSurjective f hf) I = Ideal.comap f I
Ideal.le_comap_mul 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [CommSemiring R] [CommSemiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K L : Ideal S} : Ideal.comap f K * Ideal.comap f L ≤ Ideal.comap f (K * L)
AlgHom.comap_ker 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {C : Type u_4} [Semiring C] [Algebra R C] (f : B →ₐ[R] C) (g : A →ₐ[R] B) : Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)
Ideal.comap_comapₐ 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_3} {A : Type u_4} {B : Type u_5} {C : Type u_6} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] {I : Ideal C} (f : A →ₐ[R] B) (g : B →ₐ[R] C) : Ideal.comap f (Ideal.comap g I) = Ideal.comap (g.comp f) I
Ideal.relIsoOfSurjective 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) : Ideal S ≃o { p // Ideal.comap f ⊥ ≤ p }
RingEquiv.idealComapOrderIso_apply 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (e : R ≃+* S) (I : Ideal S) : e.idealComapOrderIso I = Ideal.comap e I
Ideal.comap_symm 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} (f : R ≃+* S) : Ideal.comap f.symm I = Ideal.map f I
Ideal.map_symm 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal S} (f : R ≃+* S) : Ideal.map f.symm I = Ideal.comap f I
Ideal.disjoint_map_primeCompl_iff_comap_le 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} [CommSemiring R] {S : Type u_2} [Semiring S] {f : R →+* S} {p : Ideal R} {I : Ideal S} [p.IsPrime] : Disjoint ↑I ↑(Submonoid.map f p.primeCompl) ↔ Ideal.comap f I ≤ p
Ideal.eq_map_of_comap_eq_ker_sup 📋 Mathlib.RingTheory.Ideal.Maps
{A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] (f : A →+* B) (surj : Function.Surjective ⇑f) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap f I = RingHom.ker f ⊔ J) : I = Ideal.map f J
Ideal.exists_of_comap_eq_ker_sup 📋 Mathlib.RingTheory.Ideal.Maps
{A : Type u_1} {B : Type u_2} [Ring A] [Ring B] (f : A →+* B) (surj : Function.Surjective ⇑f) {I : Ideal B} {J : Ideal A} (eq : Ideal.comap f I = RingHom.ker f ⊔ J) {x : B} (hx : x ∈ I) : ∃ y ∈ J, f y = x
Ideal.map_comap_of_equiv 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} (f : R ≃+* S) : Ideal.map (↑f) I = Ideal.comap f.symm I
Ideal.comap_of_equiv 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} (f : R ≃+* S) : Ideal.comap (↑f) (Ideal.comap (↑f.symm) I) = I
RingEquiv.map_primeCompl_comap_eq 📋 Mathlib.RingTheory.Ideal.Maps
{R : Type u} {S : Type v} [Semiring R] [Semiring S] (e : R ≃+* S) (p : Ideal S) [p.IsPrime] : Submonoid.map e (Ideal.comap e p).primeCompl = p.primeCompl
Ideal.IsPrincipal.of_comap 📋 Mathlib.RingTheory.PrincipalIdealDomain
{R : Type u} {S : Type u_1} {F : Type u_3} [Semiring R] [Semiring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal S) [hI : Submodule.IsPrincipal (Ideal.comap f I)] : Submodule.IsPrincipal I
Ideal.polynomial_mem_ideal_of_coeff_mem_ideal 📋 Mathlib.RingTheory.Polynomial.Basic
{R : Type u} [CommSemiring R] (I : Ideal (Polynomial R)) (p : Polynomial R) (hp : ∀ (n : ℕ), p.coeff n ∈ Ideal.comap Polynomial.C I) : p ∈ I
MvPolynomial.mem_ideal_of_coeff_mem_ideal 📋 Mathlib.RingTheory.Polynomial.Basic
{R : Type u} {σ : Type v} [CommRing R] (I : Ideal (MvPolynomial σ R)) (p : MvPolynomial σ R) (hcoe : ∀ (m : σ →₀ ℕ), MvPolynomial.coeff m p ∈ Ideal.comap MvPolynomial.C I) : p ∈ I
Ideal.map_C_comap_of_comap_eq_leadingCoeff 📋 Mathlib.RingTheory.Polynomial.Basic
{R : Type u} [CommRing R] (I : Ideal (Polynomial R)) (hI : Ideal.comap Polynomial.C I = I.leadingCoeff) : Ideal.map Polynomial.C (Ideal.comap Polynomial.C I) = I
Ideal.Quotient.algebraQuotientOfLEComap 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{A : Type u_3} [Ring A] {R : Type u_5} [CommRing R] [Algebra R A] {p : Ideal R} {P : Ideal A} [P.IsTwoSided] (h : p ≤ Ideal.comap (algebraMap R A) P) : Algebra (R ⧸ p) (A ⧸ P)
Ideal.quotientAlgebra 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{A : Type u_3} [Ring A] {R : Type u_5} [CommRing R] {I : Ideal A} [I.IsTwoSided] [Algebra R A] : Algebra (R ⧸ Ideal.comap (algebraMap R A) I) (A ⧸ I)
Ideal.quotientMap 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {I : Ideal R} (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R →+* S) (hIJ : I ≤ Ideal.comap f J) : R ⧸ I →+* S ⧸ J
Ideal.instAlgebraQuotientComapRingHomAlgebraMap 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
(R : Type u_5) {A : Type u_6} [CommRing R] [CommRing A] (I : Ideal A) [Algebra R A] : Algebra (R ⧸ Ideal.comap (algebraMap R A) I) (A ⧸ I)
Ideal.quotientMapₐ 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Ring B] [Algebra R₁ B] {I : Ideal A} (J : Ideal B) [I.IsTwoSided] [J.IsTwoSided] (f : A →ₐ[R₁] B) (hIJ : I ≤ Ideal.comap f J) : A ⧸ I →ₐ[R₁] B ⧸ J
Ideal.quotientMap_comp_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : R →+* S} (H : J ≤ Ideal.comap f I) : (Ideal.quotientMap I f H).comp (Ideal.Quotient.mk J) = (Ideal.Quotient.mk I).comp f
Ideal.comap_map_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] (h : I ≤ J) : Ideal.comap (Ideal.Quotient.mk I) (Ideal.map (Ideal.Quotient.mk I) J) = J
Ideal.comap_map_quotientMk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] (I J : Ideal R) [I.IsTwoSided] : Ideal.comap (Ideal.Quotient.mk I) (Ideal.map (Ideal.Quotient.mk I) J) = I ⊔ J
Ideal.quotient_map_comp_mkₐ 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Ring B] [Algebra R₁ B] {I : Ideal A} (J : Ideal B) [I.IsTwoSided] [J.IsTwoSided] (f : A →ₐ[R₁] B) (H : I ≤ Ideal.comap f J) : (Ideal.quotientMapₐ J f H).comp (Ideal.Quotient.mkₐ R₁ I) = (Ideal.Quotient.mkₐ R₁ J).comp f
Ideal.quotientMap_surjective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : R →+* S} {H : J ≤ Ideal.comap f I} (hf : Function.Surjective ⇑f) : Function.Surjective ⇑(Ideal.quotientMap I f H)
Ideal.quotientMap_injective' 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : R →+* S} {H : J ≤ Ideal.comap f I} (h : Ideal.comap f I ≤ J) : Function.Injective ⇑(Ideal.quotientMap I f H)
Ideal.quotientMap_mk 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : R →+* S} {H : J ≤ Ideal.comap f I} {x : R} : (Ideal.quotientMap I f H) ((Ideal.Quotient.mk J) x) = (Ideal.Quotient.mk I) (f x)
Ideal.quotient_map_mkₐ 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Ring B] [Algebra R₁ B] {I : Ideal A} (J : Ideal B) [I.IsTwoSided] [J.IsTwoSided] (f : A →ₐ[R₁] B) (H : I ≤ Ideal.comap f J) {x : A} : (Ideal.quotientMapₐ J f H) ((Ideal.Quotient.mk I) x) = (Ideal.Quotient.mkₐ R₁ J) (f x)
Ideal.quotientMap_algebraMap 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R₁ : Type u_1} {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {S : Type v} [Ring S] {J : Ideal A} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided] {f : A →+* S} {H : J ≤ Ideal.comap f I} {x : R₁} : (Ideal.quotientMap I f H) ((algebraMap R₁ (A ⧸ J)) x) = (Ideal.Quotient.mk I) (f ((algebraMap R₁ A) x))
Ideal.quotientMap_injective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {I : Ideal S} {f : R →+* S} [I.IsTwoSided] : Function.Injective ⇑(Ideal.quotientMap I f ⋯)
Ideal.algebraMap_quotient_injective 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{A : Type u_3} [Ring A] {R : Type u_5} [CommRing R] {I : Ideal A} [I.IsTwoSided] [Algebra R A] : Function.Injective ⇑(algebraMap (R ⧸ Ideal.comap (algebraMap R A) I) (A ⧸ I))
Ideal.comp_quotientMap_eq_of_comp_eq 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
{R : Type u} [Ring R] {S : Type v} [Ring S] {R' : Type u_5} {S' : Type u_6} [Ring R'] [Ring S'] {f : R →+* S} {f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f) (I : Ideal S') [I.IsTwoSided] : have leq := ⋯; (Ideal.quotientMap I g' ⋯).comp (Ideal.quotientMap (Ideal.comap g' I) f ⋯) = (Ideal.quotientMap I f' ⋯).comp (Ideal.quotientMap (Ideal.comap f' I) g leq)
Ring.le_comap_jacobson 📋 Mathlib.RingTheory.Jacobson.Radical
{R : Type u_1} {R₂ : Type u_2} [Ring R] [Ring R₂] (f : R →+* R₂) [RingHomSurjective f] : Ring.jacobson R ≤ Ideal.comap f (Ring.jacobson R₂)
Ideal.comap_jacobson_of_surjective 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} (hf : Function.Surjective ⇑f) {K : Ideal S} : Ideal.comap f K.jacobson = (Ideal.comap f K).jacobson
Ideal.comap_jacobson 📋 Mathlib.RingTheory.Jacobson.Ideal
{R : Type u} {S : Type v} [Ring R] [Ring S] {f : R →+* S} {K : Ideal S} : Ideal.comap f K.jacobson = sInf (Ideal.comap f '' {J | K ≤ J ∧ J.IsMaximal})
IsLocalRing.maximalIdeal_comap 📋 Mathlib.RingTheory.LocalRing.RingHom.Basic
{R : Type u_1} {S : Type u_2} [CommSemiring R] [IsLocalRing R] [CommSemiring S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] : Ideal.comap f (IsLocalRing.maximalIdeal S) = IsLocalRing.maximalIdeal R
IsLocalRing.local_hom_TFAE 📋 Mathlib.RingTheory.LocalRing.RingHom.Basic
{R : Type u_1} {S : Type u_2} [CommSemiring R] [IsLocalRing R] [CommSemiring S] [IsLocalRing S] (f : R →+* S) : [IsLocalHom f, ⇑f '' ↑(IsLocalRing.maximalIdeal R) ⊆ ↑(IsLocalRing.maximalIdeal S), Ideal.map f (IsLocalRing.maximalIdeal R) ≤ IsLocalRing.maximalIdeal S, IsLocalRing.maximalIdeal R ≤ Ideal.comap f (IsLocalRing.maximalIdeal S), Ideal.comap f (IsLocalRing.maximalIdeal S) = IsLocalRing.maximalIdeal R].TFAE
Ideal.pointwise_smul_eq_comap 📋 Mathlib.RingTheory.Ideal.Pointwise
{M : Type u_1} {R : Type u_3} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} (S : Ideal R) : a • S = Ideal.comap (RingEquiv.symm ((MulSemiringAction.toRingAut M R) a)) S
Ideal.under_def 📋 Mathlib.RingTheory.Ideal.Over
(A : Type u_2) [CommSemiring A] {B : Type u_3} [Semiring B] [Algebra A B] (P : Ideal B) : Ideal.under A P = Ideal.comap (algebraMap A B) P
Ideal.comap_liesOver 📋 Mathlib.RingTheory.Ideal.Over
{A : Type u_2} [CommSemiring A] {B : Type u_3} {C : Type u_4} [Semiring B] [Semiring C] [Algebra A B] [Algebra A C] (Q : Ideal C) (p : Ideal A) [Q.LiesOver p] {F : Type u_6} [FunLike F B C] [AlgHomClass F A B C] (f : F) : (Ideal.comap f Q).LiesOver p
Ideal.LiesOver.of_eq_comap 📋 Mathlib.RingTheory.Ideal.Over
{A : Type u_2} [CommSemiring A] {B : Type u_3} {C : Type u_4} [Semiring B] [Semiring C] [Algebra A B] [Algebra A C] {P : Ideal B} {Q : Ideal C} (p : Ideal A) [Q.LiesOver p] {F : Type u_6} [FunLike F B C] [AlgHomClass F A B C] (f : F) (h : P = Ideal.comap f Q) : P.LiesOver p
Ideal.Quotient.algEquivOfEqComap 📋 Mathlib.RingTheory.Ideal.Over
{A : Type u_3} {B : Type u_4} {C : Type u_5} [CommRing A] [CommRing B] [CommRing C] [Algebra A B] [Algebra A C] {P : Ideal B} {Q : Ideal C} (p : Ideal A) [Q.LiesOver p] [P.LiesOver p] {E : Type u_7} [EquivLike E B C] [AlgEquivClass E A B C] (σ : E) (h : P = Ideal.comap σ Q) : (B ⧸ P) ≃ₐ[A ⧸ p] C ⧸ Q
Ideal.comap_eq_of_scalar_tower_quotient 📋 Mathlib.RingTheory.Ideal.Over
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {p : Ideal R} {P : Ideal S} [Algebra R S] [Algebra (R ⧸ p) (S ⧸ P)] [IsScalarTower R (R ⧸ p) (S ⧸ P)] (h : Function.Injective ⇑(algebraMap (R ⧸ p) (S ⧸ P))) : Ideal.comap (algebraMap R S) P = p
Ideal.Quotient.algEquivOfEqComap_apply 📋 Mathlib.RingTheory.Ideal.Over
{A : Type u_3} {B : Type u_4} {C : Type u_5} [CommRing A] [CommRing B] [CommRing C] [Algebra A B] [Algebra A C] {P : Ideal B} {Q : Ideal C} (p : Ideal A) [Q.LiesOver p] [P.LiesOver p] {E : Type u_7} [EquivLike E B C] [AlgEquivClass E A B C] (σ : E) (h : P = Ideal.comap σ Q) (x : B) : (Ideal.Quotient.algEquivOfEqComap p σ h) ((Ideal.Quotient.mk P) x) = (Ideal.Quotient.mk Q) (σ x)
Ideal.IsPrimary.comap 📋 Mathlib.RingTheory.Ideal.IsPrimary
{R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] {I : Ideal S} (hI : I.IsPrimary) (φ : R →+* S) : (Ideal.comap φ I).IsPrimary
Localization.localRingHom 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) : Localization.AtPrime I →+* Localization.AtPrime J
Localization.localRingHom_id 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] (I : Ideal R) [hI : I.IsPrime] : Localization.localRingHom I I (RingHom.id R) ⋯ = RingHom.id (Localization.AtPrime I)
Localization.localAlgHom 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] (I : Ideal S) [I.IsPrime] (J : Ideal P) [J.IsPrime] (f : S →ₐ[R] P) (hIJ : I = Ideal.comap f J) : Localization.AtPrime I →ₐ[R] Localization.AtPrime J
Localization.isLocalHom_localRingHom 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [hJ : J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) : IsLocalHom (Localization.localRingHom I J f hIJ)
Localization.le_comap_primeCompl_iff 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] {I : Ideal R} [hI : I.IsPrime] {J : Ideal P} [J.IsPrime] {f : R →+* P} : I.primeCompl ≤ Submonoid.comap f J.primeCompl ↔ Ideal.comap f J ≤ I
Localization.localAlgEquiv 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] (I : Ideal S) [I.IsPrime] (J : Ideal P) [J.IsPrime] (f : S ≃ₐ[R] P) (hIJ : I = Ideal.comap f J) : Localization.AtPrime I ≃ₐ[R] Localization.AtPrime J
Localization.localAlgHom' 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] [Algebra R P] (J : Ideal S) (K : Ideal P) [J.IsPrime] [K.IsPrime] [J.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime J)] [Localization.AtPrime.IsLiesOverAlgebra I J] [K.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime K)] [Localization.AtPrime.IsLiesOverAlgebra I K] (f : S →ₐ[R] P) (h : J = Ideal.comap f K) : Localization.AtPrime J →ₐ[Localization.AtPrime I] Localization.AtPrime K
Localization.localRingHom_comp 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] {S : Type u_4} [CommSemiring S] (J : Ideal S) [hJ : J.IsPrime] (K : Ideal P) [hK : K.IsPrime] (f : R →+* S) (hIJ : I = Ideal.comap f J) (g : S →+* P) (hJK : J = Ideal.comap g K) : Localization.localRingHom I K (g.comp f) ⋯ = (Localization.localRingHom J K g hJK).comp (Localization.localRingHom I J f hIJ)
Localization.localRingEquiv 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R ≃+* P) (hIJ : I = Ideal.comap f J) : Localization.AtPrime I ≃+* Localization.AtPrime J
Localization.localAlgEquiv' 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] [Algebra R P] (J : Ideal S) (K : Ideal P) [J.IsPrime] [K.IsPrime] [J.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime J)] [Localization.AtPrime.IsLiesOverAlgebra I J] [K.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime K)] [Localization.AtPrime.IsLiesOverAlgebra I K] (f : S ≃ₐ[R] P) (h : J = Ideal.comap f K) : Localization.AtPrime J ≃ₐ[Localization.AtPrime I] Localization.AtPrime K
Localization.localRingHom_mk 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) (x : R) (y : ↥I.primeCompl) : (Localization.localRingHom I J f hIJ) (Localization.mk x y) = Localization.mk (f x) ⟨f ↑y, ⋯⟩
Localization.localAlgHom_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] (I : Ideal S) [I.IsPrime] (J : Ideal P) [J.IsPrime] (f : S →ₐ[R] P) (hIJ : I = Ideal.comap f J) (x : Localization.AtPrime I) : (Localization.localAlgHom I J f hIJ) x = (Localization.localRingHom I J f.toRingHom hIJ) x
Localization.localRingHom_to_map 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) (x : R) : (Localization.localRingHom I J f hIJ) ((algebraMap R (Localization.AtPrime I)) x) = (algebraMap P (Localization.AtPrime J)) (f x)
Localization.localRingHom_unique 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) {j : Localization.AtPrime I →+* Localization.AtPrime J} (hj : ∀ (x : R), j ((algebraMap R (Localization.AtPrime I)) x) = (algebraMap P (Localization.AtPrime J)) (f x)) : Localization.localRingHom I J f hIJ = j
Localization.localAlgEquiv_symm_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] (I : Ideal S) [I.IsPrime] (J : Ideal P) [J.IsPrime] (f : S ≃ₐ[R] P) (hIJ : I = Ideal.comap f J) (a✝ : Localization.AtPrime J) : (Localization.localAlgEquiv I J f hIJ).symm a✝ = (Localization.localRingEquiv I J f.toRingEquiv hIJ).invFun a✝
Localization.localAlgEquiv'_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] [Algebra R P] (J : Ideal S) (K : Ideal P) [J.IsPrime] [K.IsPrime] [J.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime J)] [Localization.AtPrime.IsLiesOverAlgebra I J] [K.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime K)] [Localization.AtPrime.IsLiesOverAlgebra I K] (f : S ≃ₐ[R] P) (h : J = Ideal.comap f K) (a✝ : Localization.AtPrime J) : (Localization.localAlgEquiv' I J K f h) a✝ = (Localization.localRingHom J K (↑f) h) a✝
Localization.localAlgEquiv_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] [Algebra R P] (I : Ideal S) [I.IsPrime] (J : Ideal P) [J.IsPrime] (f : S ≃ₐ[R] P) (hIJ : I = Ideal.comap f J) (a✝ : Localization.AtPrime I) : (Localization.localAlgEquiv I J f hIJ) a✝ = (↑↑(Localization.localAlgHom I J (↑f) hIJ).toRingHom).toFun a✝
Localization.AtPrime.mapPiEvalRingHom 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] : Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I) →+* Localization.AtPrime I
Localization.localRingHom_mk' 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R →+* P) (hIJ : I = Ideal.comap f J) (x : R) (y : ↥I.primeCompl) : (Localization.localRingHom I J f hIJ) (IsLocalization.mk' (Localization.AtPrime I) x y) = IsLocalization.mk' (Localization.AtPrime J) (f x) ⟨f ↑y, ⋯⟩
Localization.localAlgHom'_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] [Algebra R P] (J : Ideal S) (K : Ideal P) [J.IsPrime] [K.IsPrime] [J.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime J)] [Localization.AtPrime.IsLiesOverAlgebra I J] [K.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime K)] [Localization.AtPrime.IsLiesOverAlgebra I K] (f : S →ₐ[R] P) (h : J = Ideal.comap f K) (a✝ : Localization.AtPrime J) : (Localization.localAlgHom' I J K f h) a✝ = ((Localization.monoidOf J.primeCompl).lift₀ ((algebraMap P (Localization.AtPrime K)).comp ↑f).toMonoidWithZeroHom ⋯) a✝
Localization.localRingEquiv_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R ≃+* P) (hIJ : I = Ideal.comap f J) (a✝ : Localization.AtPrime I) : (Localization.localRingEquiv I J f hIJ) a✝ = (↑↑(Localization.localRingHom I J (↑f) hIJ)).toFun a✝
Localization.localRingEquiv_symm_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] (J : Ideal P) [J.IsPrime] (f : R ≃+* P) (hIJ : I = Ideal.comap f J) (a : Localization.AtPrime J) : (Localization.localRingEquiv I J f hIJ).symm a = (Localization.localRingHom J I ↑f.symm ⋯) a
Localization.localAlgEquiv'_symm_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] [Algebra R S] {P : Type u_3} [CommSemiring P] (I : Ideal R) [hI : I.IsPrime] [Algebra R P] (J : Ideal S) (K : Ideal P) [J.IsPrime] [K.IsPrime] [J.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime J)] [Localization.AtPrime.IsLiesOverAlgebra I J] [K.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime K)] [Localization.AtPrime.IsLiesOverAlgebra I K] (f : S ≃ₐ[R] P) (h : J = Ideal.comap f K) (a✝ : Localization.AtPrime K) : (Localization.localAlgEquiv' I J K f h).symm a✝ = (Localization.localRingHom K J ↑{ toEquiv := ↑f.symm, map_mul' := ⋯, map_add' := ⋯ } ⋯) a✝
Localization.AtPrime.mapPiEvalRingHom_comp_algebraMap 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] : (Localization.AtPrime.mapPiEvalRingHom I).comp (algebraMap ((i : ι) → R i) (Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I))) = (algebraMap (R i) (Localization.AtPrime I)).comp (Pi.evalRingHom R i)
Localization.AtPrime.mapPiEvalRingHom_bijective 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] : Function.Bijective ⇑(Localization.AtPrime.mapPiEvalRingHom I)
Localization.AtPrime.mapPiEvalRingHom_algebraMap_apply 📋 Mathlib.RingTheory.Localization.AtPrime.Basic
{ι : Type u_4} {R : ι → Type u_5} [(i : ι) → CommSemiring (R i)] {i : ι} (I : Ideal (R i)) [I.IsPrime] {r : (i : ι) → R i} : (Localization.AtPrime.mapPiEvalRingHom I) ((algebraMap ((i : ι) → R i) (Localization.AtPrime (Ideal.comap (Pi.evalRingHom R i) I))) r) = (algebraMap (R i) (Localization.AtPrime I)) (r i)
IsLocalization.isLocalization_isLocalization_atPrime_isLocalization 📋 Mathlib.RingTheory.Localization.LocalizationLocalization
{R : Type u_1} [CommSemiring R] (M : Submonoid R) {S : Type u_2} [CommSemiring S] [Algebra R S] (T : Type u_3) [CommSemiring T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] [IsLocalization M S] (p : Ideal S) [Hp : p.IsPrime] [IsLocalization.AtPrime T p] : IsLocalization.AtPrime T (Ideal.comap (algebraMap R S) p)
IsLocalization.isLocalization_atPrime_localization_atPrime 📋 Mathlib.RingTheory.Localization.LocalizationLocalization
{R : Type u_1} [CommSemiring R] (M : Submonoid R) (p : Ideal (Localization M)) [p.IsPrime] : IsLocalization.AtPrime (Localization.AtPrime p) (Ideal.comap (algebraMap R (Localization M)) p)
IsLocalization.localizationLocalizationAtPrimeIsoLocalization 📋 Mathlib.RingTheory.Localization.LocalizationLocalization
{R : Type u_1} [CommSemiring R] (M : Submonoid R) (p : Ideal (Localization M)) [p.IsPrime] : Localization.AtPrime (Ideal.comap (algebraMap R (Localization M)) p) ≃ₐ[R] Localization.AtPrime p
RingHom.HoldsForLocalization.localRingHom 📋 Mathlib.RingTheory.LocalProperties.Basic
{P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (hPc : RingHom.StableUnderComposition fun {R S} [CommRing R] [CommRing S] => P) (hPp : RingHom.LocalizationPreserves fun {R S} [CommRing R] [CommRing S] => P) (hPl : RingHom.HoldsForLocalization fun {R S} [CommRing R] [CommRing S] => P) {R S : Type u} [CommRing R] [CommRing S] {p : Ideal R} [p.IsPrime] {q : Ideal S} [q.IsPrime] {f : R →+* S} (h : p = Ideal.comap f q) (hf : P f) : P (Localization.localRingHom p q f h)
Algebra.IsIntegral.quotient 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
{R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {I : Ideal A} [Algebra.IsIntegral R A] : Algebra.IsIntegral (R ⧸ Ideal.comap (algebraMap R A) I) (A ⧸ I)
RingHom.IsIntegral.quotient 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
{R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal S} (hf : f.IsIntegral) : (Ideal.quotientMap I f ⋯).IsIntegral
isIntegral_quotientMap_iff 📋 Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
{R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal S} : (Ideal.quotientMap I f ⋯).IsIntegral ↔ ((Ideal.Quotient.mk I).comp f).IsIntegral
Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_3} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (I : Ideal S) [I.IsMaximal] : (Ideal.comap f I).IsMaximal
Ideal.isMaximal_comap_of_isIntegral_of_isMaximal 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (I : Ideal S) [hI : I.IsMaximal] : (Ideal.comap (algebraMap R S) I).IsMaximal
Ideal.isMaximal_of_isIntegral_of_isMaximal_comap' 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (f : R →+* S) (hf : f.IsIntegral) (I : Ideal S) [I.IsPrime] (hI : (Ideal.comap f I).IsMaximal) : I.IsMaximal
Ideal.isMaximal_of_isIntegral_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (I : Ideal S) [I.IsPrime] (hI : (Ideal.comap (algebraMap R S) I).IsMaximal) : I.IsMaximal
Ideal.IntegralClosure.isMaximal_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] (I : Ideal A) [I.IsPrime] (hI : (Ideal.comap (algebraMap R A) I).IsMaximal) : I.IsMaximal
Ideal.IsIntegral.isMaximal_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] (I : Ideal A) [I.IsPrime] (hI : (Ideal.comap (algebraMap R A) I).IsMaximal) : I.IsMaximal
Ideal.IsIntegralClosure.isMaximal_of_isMaximal_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] (I : Ideal A) [I.IsPrime] (hI : (Ideal.comap (algebraMap R A) I).IsMaximal) : I.IsMaximal
Ideal.eq_bot_of_comap_eq_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {I : Ideal S} [Algebra R S] [Nontrivial R] [IsDomain S] [Algebra.IsIntegral R S] (hI : Ideal.comap (algebraMap R S) I = ⊥) : I = ⊥
Ideal.IntegralClosure.comap_ne_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
(R : Type u_1) [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [IsDomain A] [Nontrivial R] {I : Ideal A} (I_ne_bot : I ≠ ⊥) : Ideal.comap (algebraMap R A) I ≠ ⊥
Ideal.IntegralClosure.eq_bot_of_comap_eq_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
(R : Type u_1) [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [IsDomain A] [Nontrivial R] {I : Ideal A} : Ideal.comap (algebraMap R A) I = ⊥ → I = ⊥
Ideal.IsIntegral.comap_ne_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
(R : Type u_1) [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [IsDomain A] [Nontrivial R] {I : Ideal A} (I_ne_bot : I ≠ ⊥) : Ideal.comap (algebraMap R A) I ≠ ⊥
Ideal.IsIntegral.eq_bot_of_comap_eq_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
(R : Type u_1) [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [IsDomain A] [Nontrivial R] {I : Ideal A} : Ideal.comap (algebraMap R A) I = ⊥ → I = ⊥
Ideal.IsIntegralClosure.comap_ne_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
(R : Type u_1) [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [IsDomain A] [Nontrivial R] {I : Ideal A} (I_ne_bot : I ≠ ⊥) : Ideal.comap (algebraMap R A) I ≠ ⊥
Ideal.IsIntegralClosure.eq_bot_of_comap_eq_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
(R : Type u_1) [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] [IsDomain A] [Nontrivial R] {I : Ideal A} : Ideal.comap (algebraMap R A) I = ⊥ → I = ⊥
Ideal.comap_ne_bot_of_algebraic_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {I : Ideal S} [Algebra R S] [IsDomain S] {x : S} (x_ne_zero : x ≠ 0) (x_mem : x ∈ I) (hx : IsAlgebraic R x) : Ideal.comap (algebraMap R S) I ≠ ⊥
Ideal.comap_ne_bot_of_integral_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {I : Ideal S} [Algebra R S] [Nontrivial R] [IsDomain S] {x : S} (x_ne_zero : x ≠ 0) (x_mem : x ∈ I) (hx : IsIntegral R x) : Ideal.comap (algebraMap R S) I ≠ ⊥
Ideal.coeff_zero_mem_comap_of_root_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I : Ideal S} {r : S} (hr : r ∈ I) {p : Polynomial R} (hp : Polynomial.eval₂ f r p = 0) : p.coeff 0 ∈ Ideal.comap f I
Ideal.coeff_zero_mem_comap_of_root_mem_of_eval_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I : Ideal S} {r : S} (hr : r ∈ I) {p : Polynomial R} (hp : Polynomial.eval₂ f r p ∈ I) : p.coeff 0 ∈ Ideal.comap f I
Ideal.comap_ne_bot_of_root_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I : Ideal S} [IsDomain S] {r : S} (r_ne_zero : r ≠ 0) (hr : r ∈ I) {p : Polynomial R} (p_ne_zero : p ≠ 0) (hp : Polynomial.eval₂ f r p = 0) : Ideal.comap f I ≠ ⊥
Ideal.exists_ideal_over_maximal_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (P : Ideal R) [P_max : P.IsMaximal] (hP : RingHom.ker (algebraMap R S) ≤ P) : ∃ Q, Q.IsMaximal ∧ Ideal.comap (algebraMap R S) Q = P
Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomain 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [IsDomain S] [Algebra.IsIntegral R S] (P : Ideal R) [P.IsPrime] (hP : RingHom.ker (algebraMap R S) ≤ P) : ∃ Q, Q.IsPrime ∧ Ideal.comap (algebraMap R S) Q = P
Ideal.IntegralClosure.comap_lt_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] {I J : Ideal A} [I.IsPrime] (I_lt_J : I < J) : Ideal.comap (algebraMap R A) I < Ideal.comap (algebraMap R A) J
Ideal.IsIntegral.comap_lt_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] {I J : Ideal A} [I.IsPrime] (I_lt_J : I < J) : Ideal.comap (algebraMap R A) I < Ideal.comap (algebraMap R A) J
Ideal.IsIntegralClosure.comap_le_comap 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {A : Type u_3} [CommRing A] [Algebra R A] [Algebra.IsIntegral R A] {I J : Ideal A} [I.IsPrime] (I_lt_J : I < J) : Ideal.comap (algebraMap R A) I < Ideal.comap (algebraMap R A) J
Ideal.exists_coeff_ne_zero_mem_comap_of_root_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I : Ideal S} [IsDomain S] {r : S} (r_ne_zero : r ≠ 0) (hr : r ∈ I) {p : Polynomial R} : p ≠ 0 → Polynomial.eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ Ideal.comap f I
Ideal.exists_ideal_over_prime_of_isIntegral 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (P : Ideal R) [P.IsPrime] (I : Ideal S) (hIP : Ideal.comap (algebraMap R S) I ≤ P) : ∃ Q ≥ I, Q.IsPrime ∧ Ideal.comap (algebraMap R S) Q = P
Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrime 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] (P : Ideal R) [P.IsPrime] (I : Ideal S) [I.IsPrime] (hIP : Ideal.comap (algebraMap R S) I ≤ P) : ∃ Q ≥ I, Q.IsPrime ∧ Ideal.comap (algebraMap R S) Q = P
Ideal.exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I : Ideal S} {r : S} (r_non_zero_divisor : ∀ {x : S}, x * r = 0 → x = 0) (hr : r ∈ I) {p : Polynomial R} : p ≠ 0 → Polynomial.eval₂ f r p = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ Ideal.comap f I
Ideal.comap_lt_comap_of_integral_mem_sdiff 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {I J : Ideal S} [Algebra R S] [hI : I.IsPrime] (hIJ : I ≤ J) {x : S} (mem : x ∈ ↑J \ ↑I) (integral : IsIntegral R x) : Ideal.comap (algebraMap R S) I < Ideal.comap (algebraMap R S) J
Ideal.comap_lt_comap_of_root_mem_sdiff 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I J : Ideal S} [I.IsPrime] (hIJ : I ≤ J) {r : S} (hr : r ∈ ↑J \ ↑I) {p : Polynomial R} (p_ne_zero : Polynomial.map (Ideal.Quotient.mk (Ideal.comap f I)) p ≠ 0) (hp : Polynomial.eval₂ f r p ∈ I) : Ideal.comap f I < Ideal.comap f J
Ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} {I J : Ideal S} [I.IsPrime] (hIJ : I ≤ J) {r : S} (hr : r ∈ ↑J \ ↑I) {p : Polynomial R} (p_ne_zero : Polynomial.map (Ideal.Quotient.mk (Ideal.comap f I)) p ≠ 0) (hpI : Polynomial.eval₂ f r p ∈ I) : ∃ i, p.coeff i ∈ ↑(Ideal.comap f J) \ ↑(Ideal.comap f I)
Ideal.exists_nonzero_mem_of_ne_bot 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] {P : Ideal (Polynomial R)} (Pb : P ≠ ⊥) (hP : ∀ (x : R), Polynomial.C x ∈ P → x = 0) : ∃ p ∈ P, Polynomial.map (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) p ≠ 0
Ideal.injective_quotient_le_comap_map 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) : Function.Injective ⇑(Ideal.quotientMap (Ideal.map (Polynomial.mapRingHom (Ideal.Quotient.mk (Ideal.comap Polynomial.C P))) P) (Polynomial.mapRingHom (Ideal.Quotient.mk (Ideal.comap Polynomial.C P))) ⋯)
Ideal.quotient_mk_maps_eq 📋 Mathlib.RingTheory.Ideal.GoingUp
{R : Type u_1} [CommRing R] (P : Ideal (Polynomial R)) : ((Ideal.Quotient.mk (Ideal.map (Polynomial.mapRingHom (Ideal.Quotient.mk (Ideal.comap Polynomial.C P))) P)).comp Polynomial.C).comp (Ideal.Quotient.mk (Ideal.comap Polynomial.C P)) = (Ideal.quotientMap (Ideal.map (Polynomial.mapRingHom (Ideal.Quotient.mk (Ideal.comap Polynomial.C P))) P) (Polynomial.mapRingHom (Ideal.Quotient.mk (Ideal.comap Polynomial.C P))) ⋯).comp ((Ideal.Quotient.mk P).comp Polynomial.C)
RingHom.SurjectiveOnStalks.localRingHom_surjective 📋 Mathlib.RingTheory.SurjectiveOnStalks
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} (hf : f.SurjectiveOnStalks) (P : Ideal R) [P.IsPrime] (Q : Ideal S) [Q.IsPrime] (e : P = Ideal.comap f Q) : Function.Surjective ⇑(Localization.localRingHom P Q f e)
RingHom.surjectiveOnStalks_iff_forall_maximal 📋 Mathlib.RingTheory.SurjectiveOnStalks
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} : f.SurjectiveOnStalks ↔ ∀ (I : Ideal S) (x : I.IsMaximal), Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f I) I f ⋯)
RingHom.surjective_localRingHom_iff 📋 Mathlib.RingTheory.SurjectiveOnStalks
{R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] {f : R →+* S} (P : Ideal S) [P.IsPrime] : Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f P) P f ⋯) ↔ ∀ (s : S), ∃ x r, ∃ c ∉ P, f r ∉ P ∧ c * f r * s = c * f x
Ideal.ResidueField.map 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (I : Ideal R) [I.IsPrime] (J : Ideal S) [J.IsPrime] (f : R →+* S) (hf : I = Ideal.comap f J) : I.ResidueField →+* J.ResidueField
RingHom.SurjectiveOnStalks.residueFieldMap_bijective 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (H : f.SurjectiveOnStalks) (I : Ideal R) [I.IsPrime] (J : Ideal S) [J.IsPrime] (hf : I = Ideal.comap f J) : Function.Bijective ⇑(Ideal.ResidueField.map I J f hf)
Ideal.ResidueField.mapₐ 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal A) [I.IsPrime] (J : Ideal B) [J.IsPrime] (f : A →ₐ[R] B) (hf : I = Ideal.comap f.toRingHom J) : I.ResidueField →ₐ[R] J.ResidueField
Ideal.residueFieldAlgEquiv 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (J : Ideal A) (K : Ideal B) [J.IsPrime] [K.IsPrime] (f : A ≃ₐ[R] B) (h : J = Ideal.comap f K) : J.ResidueField ≃ₐ[R] K.ResidueField
Ideal.residueFieldRingEquiv 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{A : Type u_3} {B : Type u_4} [CommRing A] [CommRing B] (J : Ideal A) (K : Ideal B) [J.IsPrime] [K.IsPrime] (f : A ≃+* B) (h : J = Ideal.comap f K) : J.ResidueField ≃+* K.ResidueField
Ideal.residueFieldAlgEquiv' 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal R) [I.IsPrime] (J : Ideal A) (K : Ideal B) [J.IsPrime] [K.IsPrime] [J.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime J)] [Localization.AtPrime.IsLiesOverAlgebra I J] [K.LiesOver I] [Algebra (Localization.AtPrime I) (Localization.AtPrime K)] [Localization.AtPrime.IsLiesOverAlgebra I K] (f : A ≃ₐ[R] B) (h : J = Ideal.comap f K) : J.ResidueField ≃ₐ[I.ResidueField] K.ResidueField
Ideal.ResidueField.mapₐ_apply 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {A : Type u_3} {B : Type u_4} [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] (I : Ideal A) [I.IsPrime] (J : Ideal B) [J.IsPrime] (f : A →ₐ[R] B) (hf : I = Ideal.comap f.toRingHom J) (x : I.ResidueField) : (Ideal.ResidueField.mapₐ I J f hf) x = (Ideal.ResidueField.map I J f.toRingHom hf) x
Ideal.ResidueField.map_algebraMap 📋 Mathlib.RingTheory.LocalRing.ResidueField.Ideal
{R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] (I : Ideal R) [I.IsPrime] (J : Ideal S) [J.IsPrime] (f : R →+* S) (hf : I = Ideal.comap f J) (r : R) : (Ideal.ResidueField.map I J f hf) ((algebraMap R I.ResidueField) r) = (algebraMap S J.ResidueField) (f r)
PrimeSpectrum.comap_asIdeal 📋 Mathlib.RingTheory.Spectrum.Prime.RingHom
{R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] (f : R →+* S) (y : PrimeSpectrum S) : (PrimeSpectrum.comap f y).asIdeal = Ideal.comap f y.asIdeal
PrimeSpectrum.mem_range_comap_iff 📋 Mathlib.RingTheory.Spectrum.Prime.RingHom
{R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) {p : PrimeSpectrum R} : p ∈ Set.range (PrimeSpectrum.comap f) ↔ Ideal.comap f (Ideal.map f p.asIdeal) = p.asIdeal
image_comap_zeroLocus_eq_zeroLocus_comap 📋 Mathlib.RingTheory.Spectrum.Prime.RingHom
{R : Type u} (S : Type v) [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) (I : Ideal S) : PrimeSpectrum.comap f '' PrimeSpectrum.zeroLocus ↑I = PrimeSpectrum.zeroLocus ↑(Ideal.comap f I)
Ideal.comap_surjective_of_faithfullyFlat 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra
{A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [Module.FaithfullyFlat A B] : Function.Surjective (Ideal.comap (algebraMap A B))
Ideal.comap_map_eq_self_of_faithfullyFlat 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra
{A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [Module.FaithfullyFlat A B] (I : Ideal A) : Ideal.comap (algebraMap A B) (Ideal.map (algebraMap A B) I) = I
Ideal.minimalPrimes_comap_subset 📋 Mathlib.RingTheory.Ideal.MinimalPrime.Localization
{R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (f : R →+* S) (J : Ideal S) : (Ideal.comap f J).minimalPrimes ⊆ Ideal.comap f '' J.minimalPrimes
Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective 📋 Mathlib.RingTheory.Ideal.MinimalPrime.Localization
{R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] {f : R →+* S} (hf : Function.Injective ⇑f) (p : Ideal R) (H : p ∈ minimalPrimes R) : ∃ p', p'.IsPrime ∧ Ideal.comap f p' = p
IsLocalization.minimalPrimes_comap 📋 Mathlib.RingTheory.Ideal.MinimalPrime.Localization
{R : Type u_1} [CommSemiring R] (S : Submonoid R) (A : Type u_2) [CommSemiring A] [Algebra R A] [IsLocalization S A] (J : Ideal A) : (Ideal.comap (algebraMap R A) J).minimalPrimes = Ideal.comap (algebraMap R A) '' J.minimalPrimes

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59