Documentation

@[simp]

theorem RingHom.kerLift_mk {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) (r : R) :

f.kerLift ((Ideal.Quotient.mk (ker f)) r) = f r

theorem RingHom.lift_injective_of_ker_le_ideal {R : Type u} {S : Type v} [Ring R] [Semiring S] (I : Ideal R) [I.IsTwoSided] {f : R →+* S} (H : ∀ a ∈ I, f a = 0) (hI : ker f ≤ I) :

Function.Injective ⇑(Ideal.Quotient.lift I f H)

theorem RingHom.kerLift_injective {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) :

Function.Injective ⇑f.kerLift

The induced map from the quotient by the kernel is injective.

def RingHom.quotientKerEquivOfRightInverse {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} {g : S → R} (hf : Function.RightInverse g ⇑f) :

R ⧸ ker f ≃+* S

The first isomorphism theorem for commutative rings, computable version.

Instances For

@[simp]

theorem RingHom.quotientKerEquivOfRightInverse.apply {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} {g : S → R} (hf : Function.RightInverse g ⇑f) (x : R ⧸ ker f) :

(quotientKerEquivOfRightInverse hf) x = f.kerLift x

@[simp]

theorem RingHom.quotientKerEquivOfRightInverse.Symm.apply {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} {g : S → R} (hf : Function.RightInverse g ⇑f) (x : S) :

(quotientKerEquivOfRightInverse hf).symm x = (Ideal.Quotient.mk (ker f)) (g x)

noncomputable def RingHom.quotientKerEquivOfSurjective {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) :

R ⧸ ker f ≃+* S

The first isomorphism theorem for commutative rings, surjective case.

Instances For

@[simp]

theorem RingHom.quotientKerEquivOfSurjective_apply_mk {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (x : R) :

(quotientKerEquivOfSurjective hf) ((Ideal.Quotient.mk (ker f)) x) = f x

@[simp]

theorem RingHom.quotientKerEquivOfSurjective_symm_apply {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (x : R) :

(quotientKerEquivOfSurjective hf).symm (f x) = (Ideal.Quotient.mk (ker f)) x

theorem RingHom.quotientKerEquivOfSurjective_symm_comp {R : Type u} {S : Type v} [Ring R] [Semiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) :

(quotientKerEquivOfSurjective hf).symm.toRingHom.comp f = Ideal.Quotient.mk (ker f)

noncomputable def RingHom.quotientKerEquivRangeS {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R →+* S) :

R ⧸ ker f ≃+* ↥f.rangeS

The first isomorphism theorem for commutative rings (RingHom.rangeS version).

Instances For

noncomputable def RingHom.quotientKerEquivRange {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R →+* S) :

R ⧸ ker f ≃+* ↥f.range

The first isomorphism theorem for commutative rings (RingHom.range version).

Instances For

@[simp]

theorem Ideal.map_quotient_self {R : Type u} [Ring R] (I : Ideal R) [I.IsTwoSided] :

map (Quotient.mk I) I = ⊥

@[simp]

theorem Ideal.mk_ker {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] :

RingHom.ker (Quotient.mk I) = I

theorem Ideal.map_mk_eq_bot_of_le {R : Type u} [Ring R] {I J : Ideal R} [J.IsTwoSided] (h : I ≤ J) :

map (Quotient.mk J) I = ⊥

theorem Ideal.ker_quotient_lift {R : Type u} {S : Type v} [Ring R] [Semiring S] {I : Ideal R} [I.IsTwoSided] (f : R →+* S) (H : I ≤ RingHom.ker f) :

RingHom.ker (Quotient.lift I f H) = map (Quotient.mk I) (RingHom.ker f)

theorem Ideal.injective_lift_iff {R : Type u} {S : Type v} [Ring R] [Semiring S] {I : Ideal R} [I.IsTwoSided] {f : R →+* S} (H : ∀ a ∈ I, f a = 0) :

Function.Injective ⇑(Quotient.lift I f H) ↔ RingHom.ker f = I

theorem Ideal.ker_Pi_Quotient_mk {R : Type u} [Ring R] {ι : Type u_1} (I : ι → Ideal R) [∀ (i : ι), (I i).IsTwoSided] :

RingHom.ker (Pi.ringHom fun (i : ι) => Quotient.mk (I i)) = ⨅ (i : ι), I i

@[simp]

theorem Ideal.bot_quotient_isMaximal_iff {R : Type u} [Ring R] (I : Ideal R) [I.IsTwoSided] :

⊥.IsMaximal ↔ I.IsMaximal

@[simp]

theorem Ideal.mem_quotient_iff_mem_sup {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] {x : R} :

(Quotient.mk I) x ∈ map (Quotient.mk I) J ↔ x ∈ J ⊔ I

See also Ideal.mem_quotient_iff_mem in case I ≤ J.

theorem Ideal.mem_quotient_iff_mem {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] (hIJ : I ≤ J) {x : R} :

(Quotient.mk I) x ∈ map (Quotient.mk I) J ↔ x ∈ J

See also Ideal.mem_quotient_iff_mem_sup if the assumption I ≤ J is not available.

def Ideal.quotientInfToPiQuotient {R : Type u} [Ring R] {ι : Type u_1} (I : ι → Ideal R) [∀ (i : ι), (I i).IsTwoSided] :

R ⧸ ⨅ (i : ι), I i →+* (i : ι) → R ⧸ I i

The homomorphism from R/(⋂ i, f i) to ∏ i, (R / f i) featured in the Chinese Remainder Theorem. It is bijective if the ideals f i are coprime.

Instances For

theorem Ideal.quotientInfToPiQuotient_mk {R : Type u} [Ring R] {ι : Type u_1} (I : ι → Ideal R) [∀ (i : ι), (I i).IsTwoSided] (x : R) :

(quotientInfToPiQuotient I) ((Quotient.mk (⨅ (i : ι), I i)) x) = fun (i : ι) => (Quotient.mk (I i)) x

theorem Ideal.quotientInfToPiQuotient_mk' {R : Type u} [Ring R] {ι : Type u_1} (I : ι → Ideal R) [∀ (i : ι), (I i).IsTwoSided] (x : R) (i : ι) :

(quotientInfToPiQuotient I) ((Quotient.mk (⨅ (i : ι), I i)) x) i = (Quotient.mk (I i)) x

theorem Ideal.quotientInfToPiQuotient_inj {R : Type u} [Ring R] {ι : Type u_1} (I : ι → Ideal R) [∀ (i : ι), (I i).IsTwoSided] :

Function.Injective ⇑(quotientInfToPiQuotient I)

theorem Ideal.quotientInfToPiQuotient_surj {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hI : Pairwise (Function.onFun IsCoprime I)) :

Function.Surjective ⇑(quotientInfToPiQuotient I)

noncomputable def Ideal.quotientInfRingEquivPiQuotient {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] (f : ι → Ideal R) (hf : Pairwise (Function.onFun IsCoprime f)) :

R ⧸ ⨅ (i : ι), f i ≃+* ((i : ι) → R ⧸ f i)

Chinese Remainder Theorem. Eisenbud Ex.2.6. Similar to Atiyah-Macdonald 1.10 and Stacks 00DT

Instances For

theorem Ideal.pi_quotient_surjective {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hf : Pairwise (Function.onFun IsCoprime I)) (x : (i : ι) → R ⧸ I i) :

∃ (r : R), ∀ (i : ι), (Quotient.mk (I i)) r = x i

Corollary of Chinese Remainder Theorem: if Iᵢ are pairwise coprime ideals in a commutative ring then the canonical map R → ∏ (R ⧸ Iᵢ) is surjective.

theorem Ideal.pi_mkQ_surjective {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hI : Pairwise (Function.onFun IsCoprime I)) :

Function.Surjective ⇑(LinearMap.pi fun (i : ι) => Submodule.mkQ (I i))

theorem Ideal.exists_forall_sub_mem_ideal {R : Type u_2} [CommRing R] {ι : Type u_3} [Finite ι] {I : ι → Ideal R} (hI : Pairwise (Function.onFun IsCoprime I)) (x : ι → R) :

∃ (r : R), ∀ (i : ι), r - x i ∈ I i

Corollary of Chinese Remainder Theorem: if Iᵢ are pairwise coprime ideals in a commutative ring then given elements xᵢ you can find r with r - xᵢ ∈ Iᵢ for all i.

noncomputable def Ideal.quotientInfEquivQuotientProd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) :

R ⧸ I ⊓ J ≃+* (R ⧸ I) × R ⧸ J

Chinese remainder theorem, specialized to two ideals.

Instances For

@[simp]

theorem Ideal.quotientInfEquivQuotientProd_fst {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) (x : R ⧸ I ⊓ J) :

((I.quotientInfEquivQuotientProd J coprime) x).1 = (Quotient.factor ⋯) x

@[simp]

theorem Ideal.quotientInfEquivQuotientProd_snd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) (x : R ⧸ I ⊓ J) :

((I.quotientInfEquivQuotientProd J coprime) x).2 = (Quotient.factor ⋯) x

@[simp]

theorem Ideal.fst_comp_quotientInfEquivQuotientProd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) :

(RingHom.fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientInfEquivQuotientProd J coprime) = Quotient.factor ⋯

@[simp]

theorem Ideal.snd_comp_quotientInfEquivQuotientProd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) :

(RingHom.snd (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientInfEquivQuotientProd J coprime) = Quotient.factor ⋯

noncomputable def Ideal.quotientMulEquivQuotientProd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) :

R ⧸ I * J ≃+* (R ⧸ I) × R ⧸ J

Chinese remainder theorem, specialized to two ideals.

Instances For

@[simp]

theorem Ideal.quotientMulEquivQuotientProd_fst {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) (x : R ⧸ I * J) :

((I.quotientMulEquivQuotientProd J coprime) x).1 = (Quotient.factor ⋯) x

@[simp]

theorem Ideal.quotientMulEquivQuotientProd_snd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) (x : R ⧸ I * J) :

((I.quotientMulEquivQuotientProd J coprime) x).2 = (Quotient.factor ⋯) x

@[simp]

theorem Ideal.fst_comp_quotientMulEquivQuotientProd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) :

(RingHom.fst (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime) = Quotient.factor ⋯

@[simp]

theorem Ideal.snd_comp_quotientMulEquivQuotientProd {R : Type u_2} [CommRing R] (I J : Ideal R) (coprime : IsCoprime I J) :

(RingHom.snd (R ⧸ I) (R ⧸ J)).comp ↑(I.quotientMulEquivQuotientProd J coprime) = Quotient.factor ⋯

@[implicit_reducible]

instance Ideal.Quotient.algebra (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I : Ideal A} [I.IsTwoSided] :

Algebra R₁ (A ⧸ I)

The R₁-algebra structure on A/I for an R₁-algebra A

@[implicit_reducible]

instance Ideal.instAlgebraQuotient (R₁ : Type u_1) [CommSemiring R₁] {A : Type u_5} [CommRing A] [Algebra R₁ A] (I : Ideal A) :

Algebra R₁ (A ⧸ I)

instance Ideal.Quotient.isScalarTower (R₁ : Type u_1) (R₂ : Type u_2) {A : Type u_3} [CommSemiring R₁] [CommSemiring R₂] [Ring A] [Algebra R₁ A] [Algebra R₂ A] [SMul R₁ R₂] [IsScalarTower R₁ R₂ A] (I : Ideal A) :

IsScalarTower R₁ R₂ (A ⧸ I)

def Ideal.Quotient.mkₐ (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] :

A →ₐ[R₁] A ⧸ I

The canonical morphism A →ₐ[R₁] A ⧸ I as morphism of R₁-algebras, for I an ideal of A, where A is an R₁-algebra.

Instances For

theorem Ideal.Quotient.algHom_ext (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I : Ideal A} [I.IsTwoSided] {S : Type u_5} [Semiring S] [Algebra R₁ S] ⦃f g : A ⧸ I →ₐ[R₁] S⦄ (h : f.comp (mkₐ R₁ I) = g.comp (mkₐ R₁ I)) :

f = g

theorem Ideal.Quotient.alg_map_eq (R₁ : Type u_1) [CommSemiring R₁] {A : Type u_5} [CommRing A] [Algebra R₁ A] (I : Ideal A) :

algebraMap R₁ (A ⧸ I) = (algebraMap A (A ⧸ I)).comp (algebraMap R₁ A)

theorem Ideal.Quotient.mkₐ_toRingHom (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] :

(mkₐ R₁ I).toRingHom = mk I

@[simp]

theorem Ideal.Quotient.mkₐ_eq_mk (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] :

⇑(mkₐ R₁ I) = ⇑(mk I)

@[simp]

theorem Ideal.Quotient.algebraMap_eq {R : Type u_5} [CommRing R] (I : Ideal R) :

algebraMap R (R ⧸ I) = mk I

@[simp]

theorem Ideal.Quotient.mk_comp_algebraMap (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] :

(mk I).comp (algebraMap R₁ A) = algebraMap R₁ (A ⧸ I)

@[simp]

theorem Ideal.Quotient.mk_algebraMap (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] (x : R₁) :

(mk I) ((algebraMap R₁ A) x) = (algebraMap R₁ (A ⧸ I)) x

theorem Ideal.Quotient.mkₐ_surjective (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] :

Function.Surjective ⇑(mkₐ R₁ I)

The canonical morphism A →ₐ[R₁] I.quotient is surjective.

@[simp]

theorem Ideal.Quotient.mkₐ_ker (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] (I : Ideal A) [I.IsTwoSided] :

RingHom.ker ↑(mkₐ R₁ I) = I

The kernel of A →ₐ[R₁] I.quotient is I.

theorem Ideal.Quotient.mk_bijective_iff_eq_bot {A : Type u_3} [Ring A] (I : Ideal A) [I.IsTwoSided] :

Function.Bijective ⇑(mk I) ↔ I = ⊥

def Ideal.Quotient.factorₐ (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (hIJ : I ≤ J) :

A ⧸ I →ₐ[R₁] A ⧸ J

AlgHom version of Ideal.Quotient.factor.

Instances For

@[simp]

theorem Ideal.Quotient.coe_factorₐ (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (hIJ : I ≤ J) :

↑(factorₐ R₁ hIJ) = factor hIJ

@[simp]

theorem Ideal.Quotient.factorₐ_apply_mk (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (hIJ : I ≤ J) (x : A) :

(factorₐ R₁ hIJ) ((mk I) x) = (mk J) x

@[simp]

theorem Ideal.Quotient.factorₐ_comp_mk (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (hIJ : I ≤ J) :

(factorₐ R₁ hIJ).comp (mkₐ R₁ I) = mkₐ R₁ J

@[simp]

theorem Ideal.Quotient.factorₐ_apply (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (hIJ : I ≤ J) (x : A ⧸ I) :

(factorₐ R₁ hIJ) x = (factor hIJ) x

@[simp]

theorem Ideal.Quotient.factorₐ_refl {R : Type u_5} {A : Type u_6} [CommRing R] [CommRing A] [Algebra R A] (I : Ideal A) :

factorₐ R ⋯ = AlgHom.id R (A ⧸ I)

@[simp]

theorem Ideal.Quotient.factorₐ_comp (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (hIJ : I ≤ J) {K : Ideal A} [K.IsTwoSided] (hJK : J ≤ K) :

(factorₐ R₁ hJK).comp (factorₐ R₁ hIJ) = factorₐ R₁ ⋯

def Ideal.Quotient.liftₐ {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (I : Ideal A) [I.IsTwoSided] (f : A →ₐ[R₁] B) (hI : ∀ a ∈ I, f a = 0) :

A ⧸ I →ₐ[R₁] B

Ideal.quotient.lift as an AlgHom.

Instances For

@[simp]

theorem Ideal.Quotient.liftₐ_apply {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (I : Ideal A) [I.IsTwoSided] (f : A →ₐ[R₁] B) (hI : ∀ a ∈ I, f a = 0) (x : A ⧸ I) :

(liftₐ I f hI) x = (lift I (↑f) hI) x

theorem Ideal.Quotient.liftₐ_comp {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (I : Ideal A) [I.IsTwoSided] (f : A →ₐ[R₁] B) (hI : ∀ a ∈ I, f a = 0) :

(liftₐ I f hI).comp (mkₐ R₁ I) = f

theorem Ideal.Quotient.span_singleton_one {A : Type u_3} [Ring A] (I : Ideal A) [I.IsTwoSided] :

A ∙ 1 = ⊤

theorem Ideal.Quotient.smul_top {R : Type u_5} [CommRing R] (a : R) (I : Ideal R) :

a • ⊤ = R ∙ Submodule.Quotient.mk a

theorem Ideal.KerLift.map_smul {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) (r : R₁) (x : A ⧸ RingHom.ker f) :

f.kerLift (r • x) = r • f.kerLift x

def Ideal.kerLiftAlg {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) :

A ⧸ RingHom.ker f →ₐ[R₁] B

The induced algebras morphism from the quotient by the kernel to the codomain.

This is an isomorphism if f has a right inverse (quotientKerAlgEquivOfRightInverse) / is surjective (quotientKerAlgEquivOfSurjective).

Instances For

@[simp]

theorem Ideal.kerLiftAlg_mk {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) (a : A) :

(kerLiftAlg f) ((Quotient.mk (RingHom.ker f)) a) = f a

@[simp]

theorem Ideal.kerLiftAlg_toRingHom {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) :

↑(kerLiftAlg f) = (↑f).kerLift

theorem Ideal.kerLiftAlg_injective {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] (f : A →ₐ[R₁] B) :

Function.Injective ⇑(kerLiftAlg f)

The induced algebra morphism from the quotient by the kernel is injective.

def Ideal.quotientKerAlgEquivOfRightInverse {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} {g : B → A} (hf : Function.RightInverse g ⇑f) :

(A ⧸ RingHom.ker f) ≃ₐ[R₁] B

The first isomorphism theorem for algebras, computable version.

Instances For

@[simp]

theorem Ideal.quotientKerAlgEquivOfRightInverse_apply {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} {g : B → A} (hf : Function.RightInverse g ⇑f) (a : A ⧸ RingHom.ker f.toRingHom) :

(quotientKerAlgEquivOfRightInverse hf) a = (↑f).kerLift a

@[simp]

theorem Ideal.quotientKerAlgEquivOfRightInverse_symm_apply {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} {g : B → A} (hf : Function.RightInverse g ⇑f) (a✝ : B) :

(quotientKerAlgEquivOfRightInverse hf).symm a✝ = (Quotient.mk (RingHom.ker ↑f)) (g a✝)

noncomputable def Ideal.quotientKerAlgEquivOfSurjective {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) :

(A ⧸ RingHom.ker f) ≃ₐ[R₁] B

The first isomorphism theorem for algebras.

Instances For

theorem Ideal.quotientKerAlgEquivOfSurjective_apply {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) (a : A ⧸ RingHom.ker f.toRingHom) :

(quotientKerAlgEquivOfSurjective hf) a = (↑f).kerLift a

@[simp]

theorem Ideal.quotientKerAlgEquivOfSurjective_mk {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) (a : A) :

(quotientKerAlgEquivOfSurjective hf) ((Quotient.mk (RingHom.ker f)) a) = f a

@[simp]

theorem Ideal.quotientKerAlgEquivOfSurjective_symm_apply {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [CommSemiring R₁] [Ring A] [Algebra R₁ A] [Semiring B] [Algebra R₁ B] {f : A →ₐ[R₁] B} (hf : Function.Surjective ⇑f) (a : A) :

(quotientKerAlgEquivOfSurjective hf).symm (f a) = (Quotient.mk (RingHom.ker f)) a

noncomputable def AlgHom.liftOfSurjective {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) :

B →ₐ[R] C

AlgHom version of RingHom.liftOfSurjective that descends an algebra homomorphism along a surjection.

Instances For

@[simp]

theorem AlgHom.liftOfSurjective_apply {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) (x : A) :

(f.liftOfSurjective hf g H) (f x) = g x

theorem AlgHom.liftOfSurjective_comp {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) :

(f.liftOfSurjective hf g H).comp f = g

theorem AlgHom.liftOfSurjective_surjective {R : Type u_5} {A : Type u_6} {B : Type u_7} {C : Type u_8} [CommRing R] [CommRing A] [CommRing B] [CommRing C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (hf : Function.Surjective ⇑f) (g : A →ₐ[R] C) (H : RingHom.ker f.toRingHom ≤ RingHom.ker g.toRingHom) (hg : Function.Surjective ⇑g) :

Function.Surjective ⇑(f.liftOfSurjective hf g H)

def Ideal.quotientMap {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 ≤ comap f J) :

R ⧸ I →+* S ⧸ J

The ring hom R/I →+* S/J induced by a ring hom f : R →+* S with I ≤ f⁻¹(J)

Instances For

@[simp]

theorem Ideal.quotientMap_mk {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 ≤ comap f I} {x : R} :

(quotientMap I f H) ((Quotient.mk J) x) = (Quotient.mk I) (f x)

@[simp]

theorem Ideal.quotientMap_algebraMap {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 ≤ comap f I} {x : R₁} :

(quotientMap I f H) ((algebraMap R₁ (A ⧸ J)) x) = (Quotient.mk I) (f ((algebraMap R₁ A) x))

theorem Ideal.quotientMap_comp_mk {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 ≤ comap f I) :

(quotientMap I f H).comp (Quotient.mk J) = (Quotient.mk I).comp f

theorem Ideal.ker_quotientMap_mk {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] [J.IsTwoSided] :

RingHom.ker (quotientMap (map (Quotient.mk I) J) (Quotient.mk I) ⋯) = map (Quotient.mk J) I

def Ideal.quotientEquiv {R : Type u} [Ring R] {S : Type v} [Ring S] (I : Ideal R) (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R ≃+* S) (hIJ : J = map (↑f) I) :

R ⧸ I ≃+* S ⧸ J

The ring equiv R/I ≃+* S/J induced by a ring equiv f : R ≃+* S, where J = f(I).

Instances For

@[simp]

theorem Ideal.quotientEquiv_symm_apply {R : Type u} [Ring R] {S : Type v} [Ring S] (I : Ideal R) (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R ≃+* S) (hIJ : J = map (↑f) I) (a : S ⧸ J) :

(I.quotientEquiv J f hIJ).symm a = (quotientMap I ↑f.symm ⋯) a

@[simp]

theorem Ideal.quotientEquiv_apply {R : Type u} [Ring R] {S : Type v} [Ring S] (I : Ideal R) (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R ≃+* S) (hIJ : J = map (↑f) I) (a✝ : R ⧸ I) :

(I.quotientEquiv J f hIJ) a✝ = (↑↑(quotientMap J ↑f ⋯)).toFun a✝

theorem Ideal.quotientEquiv_mk {R : Type u} [Ring R] {S : Type v} [Ring S] (I : Ideal R) (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R ≃+* S) (hIJ : J = map (↑f) I) (x : R) :

(I.quotientEquiv J f hIJ) ((Quotient.mk I) x) = (Quotient.mk J) (f x)

theorem Ideal.quotientEquiv_symm_mk {R : Type u} [Ring R] {S : Type v} [Ring S] (I : Ideal R) (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R ≃+* S) (hIJ : J = map (↑f) I) (x : S) :

(I.quotientEquiv J f hIJ).symm ((Quotient.mk J) x) = (Quotient.mk I) (f.symm x)

theorem Ideal.quotientMap_injective' {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 ≤ comap f I} (h : comap f I ≤ J) :

Function.Injective ⇑(quotientMap I f H)

H and h are kept as separate hypothesis since H is used in constructing the quotient map.

theorem Ideal.quotientMap_injective {R : Type u} [Ring R] {S : Type v} [Ring S] {I : Ideal S} {f : R →+* S} [I.IsTwoSided] :

Function.Injective ⇑(quotientMap I f ⋯)

If we take J = I.comap f then quotientMap is injective automatically.

theorem Ideal.quotientMap_surjective {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 ≤ comap f I} (hf : Function.Surjective ⇑f) :

Function.Surjective ⇑(quotientMap I f H)

theorem Ideal.comp_quotientMap_eq_of_comp_eq {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 := ⋯; (quotientMap I g' ⋯).comp (quotientMap (comap g' I) f ⋯) = (quotientMap I f' ⋯).comp (quotientMap (comap f' I) g leq)

Commutativity of a square is preserved when taking quotients by an ideal.

def Ideal.quotientMapₐ {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 ≤ comap f J) :

A ⧸ I →ₐ[R₁] B ⧸ J

The algebra hom A/I →+* B/J induced by an algebra hom f : A →ₐ[R₁] B with I ≤ f⁻¹(J).

Instances For

@[simp]

theorem Ideal.quotient_map_mkₐ {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 ≤ comap f J) {x : A} :

(quotientMapₐ J f H) ((Quotient.mk I) x) = (Quotient.mkₐ R₁ J) (f x)

theorem Ideal.quotient_map_comp_mkₐ {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 ≤ comap f J) :

(quotientMapₐ J f H).comp (Quotient.mkₐ R₁ I) = (Quotient.mkₐ R₁ J).comp f

def Ideal.quotientEquivAlg {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 : J = map (↑f) I) :

(A ⧸ I) ≃ₐ[R₁] B ⧸ J

The algebra equiv A/I ≃ₐ[R] B/J induced by an algebra equiv f : A ≃ₐ[R] B, where J = f(I).

Instances For

@[simp]

theorem Ideal.quotientEquivAlg_symm {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 : J = map (↑f) I) :

(I.quotientEquivAlg J f hIJ).symm = J.quotientEquivAlg I f.symm ⋯

@[simp]

theorem Ideal.quotientEquivAlg_mk {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 : J = map (↑f) I) (x : A) :

(I.quotientEquivAlg J f hIJ) ((Quotient.mk I) x) = (Quotient.mk J) (f x)

@[reducible, inline]

abbrev Ideal.Quotient.algebraQuotientOfLEComap {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 ≤ comap (algebraMap R A) P) :

Algebra (R ⧸ p) (A ⧸ P)

If P lies over p, then R / p has a canonical map to A / P.

Instances For

@[implicit_reducible, instance 100]

instance Ideal.quotientAlgebra {A : Type u_3} [Ring A] {R : Type u_5} [CommRing R] {I : Ideal A} [I.IsTwoSided] [Algebra R A] :

Algebra (R ⧸ comap (algebraMap R A) I) (A ⧸ I)

@[implicit_reducible]

instance Ideal.instAlgebraQuotientComapRingHomAlgebraMap (R : Type u_5) {A : Type u_6} [CommRing R] [CommRing A] (I : Ideal A) [Algebra R A] :

Algebra (R ⧸ comap (algebraMap R A) I) (A ⧸ I)

theorem Ideal.algebraMap_quotient_injective {A : Type u_3} [Ring A] {R : Type u_5} [CommRing R] {I : Ideal A} [I.IsTwoSided] [Algebra R A] :

Function.Injective ⇑(algebraMap (R ⧸ comap (algebraMap R A) I) (A ⧸ I))

def Ideal.quotientEquivAlgOfEq (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) :

(A ⧸ I) ≃ₐ[R₁] A ⧸ J

Quotienting by equal ideals gives equivalent algebras.

Instances For

@[simp]

theorem Ideal.quotientEquivAlgOfEq_mk (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) (x : A) :

(quotientEquivAlgOfEq R₁ h) ((Quotient.mk I) x) = (Quotient.mk J) x

@[simp]

theorem Ideal.quotientEquivAlgOfEq_coe_eq_factorₐ (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) :

↑(quotientEquivAlgOfEq R₁ h) = Quotient.factorₐ R₁ ⋯

@[simp]

theorem Ideal.quotientEquivAlgOfEq_coe_eq_factor (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) :

↑(quotientEquivAlgOfEq R₁ h) = Quotient.factor ⋯

@[simp]

theorem Ideal.quotientEquivAlgOfEq_symm (R₁ : Type u_1) {A : Type u_3} [CommSemiring R₁] [Ring A] [Algebra R₁ A] {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) :

(quotientEquivAlgOfEq R₁ h).symm = quotientEquivAlgOfEq R₁ ⋯

@[simp]

theorem Ideal.comap_map_quotientMk {R : Type u} [Ring R] (I J : Ideal R) [I.IsTwoSided] :

comap (Quotient.mk I) (map (Quotient.mk I) J) = I ⊔ J

theorem Ideal.comap_map_mk {R : Type u} [Ring R] {I J : Ideal R} [I.IsTwoSided] (h : I ≤ J) :

comap (Quotient.mk I) (map (Quotient.mk I) J) = J

theorem Ideal.isPrime_map_quotientMk_of_isPrime {R : Type u} [Ring R] {I : Ideal R} [I.IsTwoSided] {p : Ideal R} [p.IsPrime] (hIP : I ≤ p) :

(map (Quotient.mk I) p).IsPrime

noncomputable def Ideal.quotientKerEquivRange {R : Type u_5} {A : Type u_6} {B : Type u_7} [CommSemiring R] [Ring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) :

(A ⧸ RingHom.ker f) ≃ₐ[R] ↥f.range

The first isomorphism theorem for commutative algebras (AlgHom.range version).

Instances For

def RingEquiv.quotientBot (R : Type u_1) [Ring R] :

R ⧸ ⊥ ≃+* R

The quotient of a ring by he zero ideal is isomorphic to the ring itself.

Instances For

@[simp]

theorem RingEquiv.quotientBot_mk {R : Type u_1} [Ring R] (r : R) :

(quotientBot R) ((Ideal.Quotient.mk ⊥) r) = r

@[simp]

theorem RingEquiv.quotientBot_symm_mk {R : Type u_1} [Ring R] (r : R) :

(quotientBot R).symm r = (Ideal.Quotient.mk ⊥) r

def AlgEquiv.quotientBot (R : Type u_1) (S : Type u_2) [CommSemiring R] [Ring S] [Algebra R S] :

(S ⧸ ⊥) ≃ₐ[R] S

RingEquiv.quotientBot as an algebra isomorphism.

Instances For

@[simp]

theorem AlgEquiv.quotientBot_mk {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommRing S] [Algebra R S] (s : S) :

(quotientBot R S) ((Ideal.Quotient.mk ⊥) s) = s

@[simp]

theorem AlgEquiv.quotientBot_symm_mk {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommRing S] [Algebra R S] (s : S) :

(quotientBot R S).symm s = (Ideal.Quotient.mk ⊥) s

def DoubleQuot.quotLeftToQuotSup {R : Type u} [CommRing R] (I J : Ideal R) :

R ⧸ I →+* R ⧸ I ⊔ J

The obvious ring hom R/I → R/(I ⊔ J)

Instances For

theorem DoubleQuot.ker_quotLeftToQuotSup {R : Type u} [CommRing R] (I J : Ideal R) :

RingHom.ker (quotLeftToQuotSup I J) = Ideal.map (Ideal.Quotient.mk I) J

The kernel of quotLeftToQuotSup

def DoubleQuot.quotQuotToQuotSup {R : Type u} [CommRing R] (I J : Ideal R) :

(R ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J →+* R ⧸ I ⊔ J

The ring homomorphism (R/I)/J' -> R/(I ⊔ J) induced by quotLeftToQuotSup where J' is the image of J in R/I

Instances For

def DoubleQuot.quotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) :

R →+* (R ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J

The composite of the maps R → (R/I) and (R/I) → (R/I)/J'

Instances For

theorem DoubleQuot.ker_quotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) :

RingHom.ker (quotQuotMk I J) = I ⊔ J

The kernel of quotQuotMk

def DoubleQuot.liftSupQuotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) :

R ⧸ I ⊔ J →+* (R ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J

The ring homomorphism R/(I ⊔ J) → (R/I)/J' induced by quotQuotMk

Instances For

def DoubleQuot.quotQuotEquivQuotSup {R : Type u} [CommRing R] (I J : Ideal R) :

(R ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J ≃+* R ⧸ I ⊔ J

quotQuotToQuotSup and liftSupQuotQuotMk are inverse isomorphisms. In the case where I ≤ J, this is the Third Isomorphism Theorem (see quotQuotEquivQuotOfLe).

Instances For

@[simp]

theorem DoubleQuot.quotQuotEquivQuotSup_quotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) (x : R) :

(quotQuotEquivQuotSup I J) ((quotQuotMk I J) x) = (Ideal.Quotient.mk (I ⊔ J)) x

@[simp]

theorem DoubleQuot.quotQuotEquivQuotSup_symm_quotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) (x : R) :

(quotQuotEquivQuotSup I J).symm ((Ideal.Quotient.mk (I ⊔ J)) x) = (quotQuotMk I J) x

def DoubleQuot.quotQuotEquivComm {R : Type u} [CommRing R] (I J : Ideal R) :

(R ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J ≃+* (R ⧸ J) ⧸ Ideal.map (Ideal.Quotient.mk J) I

The obvious isomorphism (R/I)/J' → (R/J)/I'

Instances For

@[simp]

theorem DoubleQuot.quotQuotEquivComm_quotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) (x : R) :

(quotQuotEquivComm I J) ((quotQuotMk I J) x) = (quotQuotMk J I) x

@[simp]

theorem DoubleQuot.quotQuotEquivComm_comp_quotQuotMk {R : Type u} [CommRing R] (I J : Ideal R) :

(↑(quotQuotEquivComm I J)).comp (quotQuotMk I J) = quotQuotMk J I

@[simp]

theorem DoubleQuot.quotQuotEquivComm_symm {R : Type u} [CommRing R] (I J : Ideal R) :

(quotQuotEquivComm I J).symm = quotQuotEquivComm J I

def DoubleQuot.quotQuotEquivQuotOfLE {R : Type u} [CommRing R] {I J : Ideal R} (h : I ≤ J) :

(R ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J ≃+* R ⧸ J

The Third Isomorphism theorem for rings. See quotQuotEquivQuotSup for a version that does not assume an inclusion of ideals.

Instances For

@[simp]

theorem DoubleQuot.quotQuotEquivQuotOfLE_quotQuotMk {R : Type u} [CommRing R] {I J : Ideal R} (x : R) (h : I ≤ J) :

(quotQuotEquivQuotOfLE h) ((quotQuotMk I J) x) = (Ideal.Quotient.mk J) x

@[simp]

theorem DoubleQuot.quotQuotEquivQuotOfLE_symm_mk {R : Type u} [CommRing R] {I J : Ideal R} (x : R) (h : I ≤ J) :

(quotQuotEquivQuotOfLE h).symm ((Ideal.Quotient.mk J) x) = (quotQuotMk I J) x

theorem DoubleQuot.quotQuotEquivQuotOfLE_comp_quotQuotMk {R : Type u} [CommRing R] {I J : Ideal R} (h : I ≤ J) :

(↑(quotQuotEquivQuotOfLE h)).comp (quotQuotMk I J) = Ideal.Quotient.mk J

theorem DoubleQuot.quotQuotEquivQuotOfLE_symm_comp_mk {R : Type u} [CommRing R] {I J : Ideal R} (h : I ≤ J) :

(↑(quotQuotEquivQuotOfLE h).symm).comp (Ideal.Quotient.mk J) = quotQuotMk I J

@[simp]

theorem DoubleQuot.quotQuotEquivComm_mk_mk {R : Type u} [CommRing R] (I J : Ideal R) (x : R) :

(quotQuotEquivComm I J) ((Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk I) J)) ((Ideal.Quotient.mk I) x)) = (algebraMap R ((R ⧸ J) ⧸ Ideal.map (Ideal.Quotient.mk J) I)) x

@[simp]

theorem DoubleQuot.quotQuotEquivQuotSup_quot_quot_algebraMap {R : Type u} [CommSemiring R] {A : Type v} [CommRing A] [Algebra R A] (I J : Ideal A) (x : R) :

(quotQuotEquivQuotSup I J) ((algebraMap R ((A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)) x) = (algebraMap R (A ⧸ I ⊔ J)) x

@[simp]

theorem DoubleQuot.quotQuotEquivComm_algebraMap {R : Type u} [CommSemiring R] {A : Type v} [CommRing A] [Algebra R A] (I J : Ideal A) (x : R) :

(quotQuotEquivComm I J) ((algebraMap R ((A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mk I) J)) x) = (algebraMap R ((A ⧸ J) ⧸ Ideal.map (Ideal.Quotient.mk J) I)) x

def DoubleQuot.quotLeftToQuotSupₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

A ⧸ I →ₐ[R] A ⧸ I ⊔ J

The natural algebra homomorphism A / I → A / (I ⊔ J).

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@[simp]

theorem DoubleQuot.quotLeftToQuotSupₐ_toRingHom (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(quotLeftToQuotSupₐ R I J) = quotLeftToQuotSup I J

@[simp]

theorem DoubleQuot.coe_quotLeftToQuotSupₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(quotLeftToQuotSupₐ R I J) = ⇑(quotLeftToQuotSup I J)

def DoubleQuot.quotQuotToQuotSupₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

(A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mkₐ R I) J →ₐ[R] A ⧸ I ⊔ J

The algebra homomorphism (A / I) / J' -> A / (I ⊔ J) induced by quotQuotToQuotSup, where J' is the projection of J in A / I.

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@[simp]

theorem DoubleQuot.quotQuotToQuotSupₐ_toRingHom (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(quotQuotToQuotSupₐ R I J) = quotQuotToQuotSup I J

@[simp]

theorem DoubleQuot.coe_quotQuotToQuotSupₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(quotQuotToQuotSupₐ R I J) = ⇑(quotQuotToQuotSup I J)

def DoubleQuot.quotQuotMkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

A →ₐ[R] (A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mkₐ R I) J

The composition of the algebra homomorphisms A → (A / I) and (A / I) → (A / I) / J', where J' is the projection J in A / I.

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@[simp]

theorem DoubleQuot.quotQuotMkₐ_toRingHom (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(quotQuotMkₐ R I J) = quotQuotMk I J

@[simp]

theorem DoubleQuot.coe_quotQuotMkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(quotQuotMkₐ R I J) = ⇑(quotQuotMk I J)

def DoubleQuot.liftSupQuotQuotMkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

A ⧸ I ⊔ J →ₐ[R] (A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mkₐ R I) J

The injective algebra homomorphism A / (I ⊔ J) → (A / I) / J' induced by quotQuotMk, where J' is the projection J in A / I.

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@[simp]

theorem DoubleQuot.liftSupQuotQuotMkₐ_toRingHom (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(liftSupQuotQuotMkₐ R I J) = liftSupQuotQuotMk I J

@[simp]

theorem DoubleQuot.coe_liftSupQuotQuotMkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(liftSupQuotQuotMkₐ R I J) = ⇑(liftSupQuotQuotMk I J)

def DoubleQuot.quotQuotEquivQuotSupₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

((A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mkₐ R I) J) ≃ₐ[R] A ⧸ I ⊔ J

quotQuotToQuotSup and liftSupQuotQuotMk are inverse isomorphisms. In the case where I ≤ J, this is the Third Isomorphism Theorem (see DoubleQuot.quotQuotEquivQuotOfLE).

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@[simp]

theorem DoubleQuot.quotQuotEquivQuotSupₐ_toRingEquiv (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(quotQuotEquivQuotSupₐ R I J) = quotQuotEquivQuotSup I J

@[simp]

theorem DoubleQuot.coe_quotQuotEquivQuotSupₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(quotQuotEquivQuotSupₐ R I J) = ⇑(quotQuotEquivQuotSup I J)

@[simp]

theorem DoubleQuot.quotQuotEquivQuotSupₐ_symm_toRingEquiv (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(quotQuotEquivQuotSupₐ R I J).symm = (quotQuotEquivQuotSup I J).symm

@[simp]

theorem DoubleQuot.coe_quotQuotEquivQuotSupₐ_symm (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(quotQuotEquivQuotSupₐ R I J).symm = ⇑(quotQuotEquivQuotSup I J).symm

def DoubleQuot.quotQuotEquivCommₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

((A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mkₐ R I) J) ≃ₐ[R] (A ⧸ J) ⧸ Ideal.map (Ideal.Quotient.mkₐ R J) I

The natural algebra isomorphism (A / I) / J' → (A / J) / I', where J' (resp. I') is the projection of J in A / I (resp. I in A / J).

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@[simp]

theorem DoubleQuot.quotQuotEquivCommₐ_toRingEquiv (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

↑(quotQuotEquivCommₐ R I J) = quotQuotEquivComm I J

@[simp]

theorem DoubleQuot.coe_quotQuotEquivCommₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

⇑(quotQuotEquivCommₐ R I J) = ⇑(quotQuotEquivComm I J)

@[simp]

theorem DoubleQuot.quotQuotEquivComm_symmₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

(quotQuotEquivCommₐ R I J).symm = quotQuotEquivCommₐ R J I

@[simp]

theorem DoubleQuot.quotQuotEquivComm_comp_quotQuotMkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] (I J : Ideal A) :

(↑(quotQuotEquivCommₐ R I J)).comp (quotQuotMkₐ R I J) = quotQuotMkₐ R J I

def DoubleQuot.quotQuotEquivQuotOfLEₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

((A ⧸ I) ⧸ Ideal.map (Ideal.Quotient.mkₐ R I) J) ≃ₐ[R] A ⧸ J

The third isomorphism theorem for algebras. See quotQuotEquivQuotSupₐ for version that does not assume an inclusion of ideals.

Instances For

@[simp]

theorem DoubleQuot.quotQuotEquivQuotOfLEₐ_toRingEquiv (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

↑(quotQuotEquivQuotOfLEₐ R h) = quotQuotEquivQuotOfLE h

@[simp]

theorem DoubleQuot.coe_quotQuotEquivQuotOfLEₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

⇑(quotQuotEquivQuotOfLEₐ R h) = ⇑(quotQuotEquivQuotOfLE h)

@[simp]

theorem DoubleQuot.quotQuotEquivQuotOfLEₐ_symm_toRingEquiv (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

↑(quotQuotEquivQuotOfLEₐ R h).symm = (quotQuotEquivQuotOfLE h).symm

@[simp]

theorem DoubleQuot.coe_quotQuotEquivQuotOfLEₐ_symm (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

⇑(quotQuotEquivQuotOfLEₐ R h).symm = ⇑(quotQuotEquivQuotOfLE h).symm

@[simp]

theorem DoubleQuot.quotQuotEquivQuotOfLE_comp_quotQuotMkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

(↑(quotQuotEquivQuotOfLEₐ R h)).comp (quotQuotMkₐ R I J) = Ideal.Quotient.mkₐ R J

@[simp]

theorem DoubleQuot.quotQuotEquivQuotOfLE_symm_comp_mkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

(↑(quotQuotEquivQuotOfLEₐ R h).symm).comp (Ideal.Quotient.mkₐ R J) = quotQuotMkₐ R I J

theorem DoubleQuot.quotQuotEquivQuotOfLEₐ_comp_mkₐ (R : Type u) {A : Type u_1} [CommSemiring R] [CommRing A] [Algebra R A] {I J : Ideal A} (h : I ≤ J) :

(↑(quotQuotEquivQuotOfLEₐ R h)).comp (Ideal.Quotient.mkₐ R (Ideal.map (Ideal.Quotient.mkₐ R I) J)) = Ideal.Quotient.factorₐ R h

noncomputable def Ideal.powQuotPowSuccLinearEquivMapMkPowSuccPow {R : Type u_1} [CommRing R] (I : Ideal R) (n : ℕ) :

(↥(I ^ n) ⧸ I • ⊤) ≃ₗ[R] ↥(map (Quotient.mk (I ^ (n + 1))) (I ^ n))

I ^ n ⧸ I ^ (n + 1) can be viewed as a quotient module and as ideal of R ⧸ I ^ (n + 1). This definition gives the R-linear equivalence between the two.

Instances For