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ring_theory.ideal.quotient_operations

More operations on modules and ideals related to quotients #

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def ring_hom.ker_lift {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) :

The induced map from the quotient by the kernel to the codomain.

This is an isomorphism if f has a right inverse (quotient_ker_equiv_of_right_inverse) / is surjective (quotient_ker_equiv_of_surjective).

Equations
@[simp]
theorem ring_hom.ker_lift_mk {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) (r : R) :
theorem ring_hom.ker_lift_injective {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) :

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

theorem ring_hom.lift_injective_of_ker_le_ideal {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (I : ideal R) {f : R →+* S} (H : (a : R), a I f a = 0) (hI : ring_hom.ker f I) :

The first isomorphism theorem for commutative rings, computable version.

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noncomputable def ring_hom.quotient_ker_equiv_of_surjective {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {f : R →+* S} (hf : function.surjective f) :

The first isomorphism theorem for commutative rings.

Equations
@[simp]
@[simp]
theorem ideal.mk_ker {R : Type u} [comm_ring R] {I : ideal R} :
theorem ideal.map_mk_eq_bot_of_le {R : Type u} [comm_ring R] {I J : ideal R} (h : I J) :
@[simp]
theorem ideal.mem_quotient_iff_mem_sup {R : Type u} [comm_ring R] {I J : ideal R} {x : R} :

See also ideal.mem_quotient_iff_mem in case I ≤ J.

theorem ideal.mem_quotient_iff_mem {R : Type u} [comm_ring R] {I J : ideal R} (hIJ : I J) {x : R} :

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

theorem ideal.comap_map_mk {R : Type u} [comm_ring R] {I J : ideal R} (h : I J) :
@[protected, instance]
def ideal.quotient.algebra (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] {I : ideal A} :
algebra R₁ (A I)

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

Equations
@[protected, instance]
def ideal.quotient.is_scalar_tower (R₁ : Type u_1) (R₂ : Type u_2) {A : Type u_3} [comm_semiring R₁] [comm_semiring R₂] [comm_ring A] [algebra R₁ A] [algebra R₂ A] [has_smul R₁ R₂] [is_scalar_tower R₁ R₂ A] (I : ideal A) :
is_scalar_tower R₁ R₂ (A I)
def ideal.quotient.mkₐ (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :
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.

Equations
theorem ideal.quotient.alg_hom_ext (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] {I : ideal A} {S : Type u_2} [semiring S] [algebra R₁ S] ⦃f g : A I →ₐ[R₁] S⦄ (h : f.comp (ideal.quotient.mkₐ R₁ I) = g.comp (ideal.quotient.mkₐ R₁ I)) :
f = g
theorem ideal.quotient.alg_map_eq (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :
algebra_map R₁ (A I) = (algebra_map A (A I)).comp (algebra_map R₁ A)
theorem ideal.quotient.mkₐ_to_ring_hom (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :
@[simp]
theorem ideal.quotient.mkₐ_eq_mk (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :
@[simp]
@[simp]
theorem ideal.quotient.mk_comp_algebra_map (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :
@[simp]
theorem ideal.quotient.mk_algebra_map (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) (x : R₁) :
(ideal.quotient.mk I) ((algebra_map R₁ A) x) = (algebra_map R₁ (A I)) x
theorem ideal.quotient.mkₐ_surjective (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :

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

@[simp]
theorem ideal.quotient.mkₐ_ker (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] (I : ideal A) :

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

def ideal.quotient.liftₐ {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (I : ideal A) (f : A →ₐ[R₁] B) (hI : (a : A), a I f a = 0) :
A I →ₐ[R₁] B

ideal.quotient.lift as an alg_hom.

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@[simp]
theorem ideal.quotient.liftₐ_apply {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (I : ideal A) (f : A →ₐ[R₁] B) (hI : (a : A), a I f a = 0) (x : A I) :
theorem ideal.quotient.liftₐ_comp {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (I : ideal A) (f : A →ₐ[R₁] B) (hI : (a : A), a I f a = 0) :
theorem ideal.ker_lift.map_smul {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (f : A →ₐ[R₁] B) (r : R₁) (x : A ring_hom.ker f.to_ring_hom) :
def ideal.ker_lift_alg {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (f : A →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 (quotient_ker_alg_equiv_of_right_inverse) / is surjective (quotient_ker_alg_equiv_of_surjective).

Equations
@[simp]
theorem ideal.ker_lift_alg_mk {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (f : A →ₐ[R₁] B) (a : A) :
@[simp]
theorem ideal.ker_lift_alg_to_ring_hom {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (f : A →ₐ[R₁] B) :
theorem ideal.ker_lift_alg_injective {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (f : A →ₐ[R₁] B) :

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

def ideal.quotient_ker_alg_equiv_of_right_inverse {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] {f : A →ₐ[R₁] B} {g : B A} (hf : function.right_inverse g f) :

The first isomorphism theorem for algebras, computable version.

Equations
@[simp]
noncomputable def ideal.quotient_ker_alg_equiv_of_surjective {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] {f : A →ₐ[R₁] B} (hf : function.surjective f) :

The first isomorphism theorem for algebras.

Equations
def ideal.quotient_map {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {I : ideal R} (J : ideal S) (f : R →+* S) (hIJ : I ideal.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)

Equations
@[simp]
theorem ideal.quotient_map_mk {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ideal.comap f I} {x : R} :
@[simp]
theorem ideal.quotient_map_algebra_map {S : Type v} [comm_ring S] {R₁ : Type u_1} {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] {J : ideal A} {I : ideal S} {f : A →+* S} {H : J ideal.comap f I} {x : R₁} :
(I.quotient_map f H) ((algebra_map R₁ (A J)) x) = (ideal.quotient.mk I) (f ((algebra_map R₁ A) x))
theorem ideal.quotient_map_comp_mk {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} (H : J ideal.comap f I) :
@[simp]
theorem ideal.quotient_equiv_symm_apply {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map f I) (ᾰ : S J) :
((I.quotient_equiv J f hIJ).symm) = (I.quotient_map (f.symm) _)
def ideal.quotient_equiv {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.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).

Equations
@[simp]
theorem ideal.quotient_equiv_apply {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map f I) (ᾰ : R I) :
(I.quotient_equiv J f hIJ) = (J.quotient_map f _).to_fun
@[simp]
theorem ideal.quotient_equiv_mk {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map f I) (x : R) :
@[simp]
theorem ideal.quotient_equiv_symm_mk {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (I : ideal R) (J : ideal S) (f : R ≃+* S) (hIJ : J = ideal.map f I) (x : S) :
theorem ideal.quotient_map_injective' {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ideal.comap f I} (h : ideal.comap f I J) :

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

theorem ideal.quotient_map_injective {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {I : ideal S} {f : R →+* S} :

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

theorem ideal.quotient_map_surjective {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ideal.comap f I} (hf : function.surjective f) :
theorem ideal.comp_quotient_map_eq_of_comp_eq {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {R' : Type u_1} {S' : Type u_2} [comm_ring R'] [comm_ring S'] {f : R →+* S} {f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f) (I : ideal S') :

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

def ideal.quotient_mapₐ {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] {I : ideal A} (J : ideal B) (f : A →ₐ[R₁] B) (hIJ : I ideal.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).

Equations
@[simp]
theorem ideal.quotient_map_mkₐ {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] {I : ideal A} (J : ideal B) (f : A →ₐ[R₁] B) (H : I ideal.comap f J) {x : A} :
theorem ideal.quotient_map_comp_mkₐ {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] {I : ideal A} (J : ideal B) (f : A →ₐ[R₁] B) (H : I ideal.comap f J) :
def ideal.quotient_equiv_alg {R₁ : Type u_1} {A : Type u_3} {B : Type u_4} [comm_semiring R₁] [comm_ring A] [comm_ring B] [algebra R₁ A] [algebra R₁ B] (I : ideal A) (J : ideal B) (f : A ≃ₐ[R₁] B) (hIJ : J = ideal.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, whereJ = f(I).

Equations
@[protected, instance]
def ideal.quotient_algebra {R : Type u} [comm_ring R] {A : Type u_3} [comm_ring A] {I : ideal A} [algebra R A] :
Equations
def ideal.quotient_equiv_alg_of_eq (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] {I J : ideal A} (h : I = J) :
(A I) ≃ₐ[R₁] A J

Quotienting by equal ideals gives equivalent algebras.

Equations
@[simp]
theorem ideal.quotient_equiv_alg_of_eq_mk (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] {I J : ideal A} (h : I = J) (x : A) :
@[simp]
theorem ideal.quotient_equiv_alg_of_eq_symm (R₁ : Type u_1) {A : Type u_3} [comm_semiring R₁] [comm_ring A] [algebra R₁ A] {I J : ideal A} (h : I = J) :
def double_quot.quot_left_to_quot_sup {R : Type u} [comm_ring R] (I J : ideal R) :
R I →+* R I J

The obvious ring hom R/I → R/(I ⊔ J)

Equations

The kernel of quot_left_to_quot_sup

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

Equations
def double_quot.quot_quot_mk {R : Type u} [comm_ring R] (I J : ideal R) :

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

Equations

The kernel of quot_quot_mk

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

Equations

quot_quot_to_quot_add and lift_sup_double_qot_mk are inverse isomorphisms. In the case where I ≤ J, this is the Third Isomorphism Theorem (see quot_quot_equiv_quot_of_le)

Equations
def double_quot.quot_quot_equiv_quot_of_le {R : Type u} [comm_ring R] {I J : ideal R} (h : I J) :

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

Equations
def double_quot.quot_left_to_quot_supₐ (R : Type u) {A : Type u_1} [comm_semiring R] [comm_ring A] [algebra R A] (I J : ideal A) :
A I →ₐ[R] A I J

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

Equations
def double_quot.quot_quot_to_quot_supₐ (R : Type u) {A : Type u_1} [comm_semiring R] [comm_ring A] [algebra R A] (I J : ideal A) :

The algebra homomorphism (A / I) / J' -> A / (I ⊔ J) induced byquot_left_to_quot_sup, whereJ'is the projection ofJinA / I`.

Equations
def double_quot.quot_quot_mkₐ (R : Type u) {A : Type u_1} [comm_semiring R] [comm_ring A] [algebra R A] (I J : ideal A) :

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

Equations
def double_quot.lift_sup_quot_quot_mkₐ (R : Type u) {A : Type u_1} [comm_semiring R] [comm_ring A] [algebra R A] (I J : ideal A) :

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

Equations
def double_quot.quot_quot_equiv_quot_supₐ (R : Type u) {A : Type u_1} [comm_semiring R] [comm_ring A] [algebra R A] (I J : ideal A) :

quot_quot_to_quot_add and lift_sup_quot_quot_mk are inverse isomorphisms. In the case where I ≤ J, this is the Third Isomorphism Theorem (see quot_quot_equiv_quot_of_le).

Equations

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).

Equations
def double_quot.quot_quot_equiv_quot_of_leₐ (R : Type u) {A : Type u_1} [comm_semiring R] [comm_ring A] [algebra R A] {I J : ideal A} (h : I J) :

The third isomoprhism theorem for rings. See quot_quot_equiv_quot_sup for version that does not assume an inclusion of ideals.

Equations