Documentation

Mathlib.RingTheory.Ideal.Maps

Maps on modules and ideals #

Main definitions include Ideal.map, Ideal.comap, RingHom.ker, Module.annihilator and Submodule.annihilator.

def Ideal.map {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (I : Ideal R) :

I.map f is the span of the image of the ideal I under f, which may be bigger than the image itself.

Equations
def Ideal.comap {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) :

I.comap f is the preimage of I under f.

Equations
  • Ideal.comap f I = { carrier := f ⁻¹' I, add_mem' := , zero_mem' := , smul_mem' := }
@[simp]
theorem Ideal.coe_comap {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
theorem Ideal.comap_coe {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) :
theorem Ideal.map_mono {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {I : Ideal R} {J : Ideal R} (h : I J) :
theorem Ideal.mem_map_of_mem {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {I : Ideal R} {x : R} (h : x I) :
f x Ideal.map f I
theorem Ideal.apply_coe_mem_map {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (I : Ideal R) (x : I) :
f x Ideal.map f I
theorem Ideal.map_le_iff_le_comap {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] :
@[simp]
theorem Ideal.mem_comap {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
theorem Ideal.comap_mono {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} {K : Ideal S} {L : Ideal S} [RingHomClass F R S] (h : K L) :
theorem Ideal.comap_ne_top {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 ) :
theorem Ideal.map_le_comap_of_inv_on {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) :
theorem Ideal.comap_le_map_of_inv_on {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)) :
theorem Ideal.map_le_comap_of_inverse {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) :

The Ideal version of Set.image_subset_preimage_of_inverse.

theorem Ideal.comap_le_map_of_inverse {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) :

The Ideal version of Set.preimage_subset_image_of_inverse.

instance Ideal.IsPrime.comap {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
Equations
  • =
theorem Ideal.map_top {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] :
theorem Ideal.gc_map_comap {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) [RingHomClass F R S] :
@[simp]
theorem Ideal.comap_id {R : Type u} [Semiring R] (I : Ideal R) :
@[simp]
theorem Ideal.map_id {R : Type u} [Semiring R] (I : Ideal R) :
theorem Ideal.comap_comap {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
theorem Ideal.comap_comapₐ {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
theorem Ideal.map_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] {T : Type u_3} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) :
Ideal.map g (Ideal.map f I) = Ideal.map (g.comp f) I
theorem Ideal.map_mapₐ {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 A} (f : A →ₐ[R] B) (g : B →ₐ[R] C) :
Ideal.map g (Ideal.map f I) = Ideal.map (g.comp f) I
theorem Ideal.map_span {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [RingHomClass F R S] (f : F) (s : Set R) :
theorem Ideal.map_le_of_le_comap {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 KIdeal.map f I K
theorem Ideal.le_comap_of_map_le {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 KI Ideal.comap f K
theorem Ideal.le_comap_map {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] :
theorem Ideal.map_comap_le {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] :
@[simp]
theorem Ideal.comap_top {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} [RingHomClass F R S] :
@[simp]
theorem Ideal.comap_eq_top_iff {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} :
@[simp]
theorem Ideal.map_bot {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] {f : F} [RingHomClass F R S] :
theorem Ideal.ne_bot_of_map_ne_bot {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] (hI : Ideal.map f I ) :
@[simp]
theorem Ideal.map_comap_map {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] :
@[simp]
theorem Ideal.comap_map_comap {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] :
theorem Ideal.map_sup {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (I : Ideal R) (J : Ideal R) [RingHomClass F R S] :
theorem Ideal.comap_inf {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) (K : Ideal S) (L : Ideal S) [RingHomClass F R S] :
theorem Ideal.map_iSup {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 : ιIdeal R) :
Ideal.map f (iSup K) = ⨆ (i : ι), Ideal.map f (K i)
theorem Ideal.comap_iInf {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 : ιIdeal S) :
Ideal.comap f (iInf K) = ⨅ (i : ι), Ideal.comap f (K i)
theorem Ideal.map_sSup {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 R)) :
Ideal.map f (sSup s) = Is, Ideal.map f I
theorem Ideal.comap_sInf {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) = Is, Ideal.comap f I
theorem Ideal.comap_sInf' {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) = IIdeal.comap f '' s, I
theorem Ideal.comap_isPrime {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

Variant of Ideal.IsPrime.comap where ideal is explicit rather than implicit.

theorem Ideal.map_inf_le {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {I : Ideal R} {J : Ideal R} [RingHomClass F R S] :
theorem Ideal.le_comap_sup {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] (f : F) {K : Ideal S} {L : Ideal S} [RingHomClass F R S] :
theorem element_smul_restrictScalars {R : Type u_4} {S : Type u_5} {M : Type u_6} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] (r : R) (N : Submodule S M) :
@[simp]
@[simp]
theorem Ideal.coe_restrictScalars {R : Type u_4} {S : Type u_5} [CommSemiring R] [Semiring S] [Algebra R S] (I : Ideal S) :
@[simp]

The smallest S-submodule that contains all x ∈ I * y ∈ J is also the smallest R-submodule that does so.

theorem Ideal.map_comap_of_surjective {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) :
def Ideal.giMapComap {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) :

map and comap are adjoint, and the composition map f ∘ comap f is the identity

Equations
theorem Ideal.map_surjective_of_surjective {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) :
theorem Ideal.comap_injective_of_surjective {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) :
theorem Ideal.map_sup_comap_of_surjective {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) (J : Ideal S) :
theorem Ideal.map_iSup_comap_of_surjective {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 : ιIdeal S) :
Ideal.map f (⨆ (i : ι), Ideal.comap f (K i)) = iSup K
theorem Ideal.map_inf_comap_of_surjective {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) (J : Ideal S) :
theorem Ideal.map_iInf_comap_of_surjective {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 : ιIdeal S) :
Ideal.map f (⨅ (i : ι), Ideal.comap f (K i)) = iInf K
theorem Ideal.mem_image_of_mem_map_of_surjective {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 R} {y : S} (H : y Ideal.map f I) :
y f '' I
theorem Ideal.mem_map_iff_of_surjective {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 R} {y : S} :
y Ideal.map f I xI, f x = y
theorem Ideal.le_map_of_comap_le_of_surjective {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 IK Ideal.map f I
theorem Ideal.map_eq_submodule_map {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R →+* S) [h : RingHomSurjective f] (I : Ideal R) :
Ideal.map f I = Submodule.map f.toSemilinearMap I
theorem Ideal.comap_bot_le_of_injective {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) :
theorem Ideal.comap_bot_of_injective {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) :
@[simp]
theorem Ideal.map_of_equiv {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} (f : R ≃+* S) :
Ideal.map (↑f.symm) (Ideal.map (↑f) I) = I

If f : R ≃+* S is a ring isomorphism and I : Ideal R, then map f.symm (map f I) = I.

@[simp]
theorem Ideal.comap_of_equiv {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

If f : R ≃+* S is a ring isomorphism and I : Ideal R, then comap f (comap f.symm I) = I.

theorem Ideal.map_comap_of_equiv {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

If f : R ≃+* S is a ring isomorphism and I : Ideal R, then map f I = comap f.symm I.

@[simp]
theorem Ideal.comap_symm {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

If f : R ≃+* S is a ring isomorphism and I : Ideal R, then comap f.symm I = map f I.

@[simp]
theorem Ideal.map_symm {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

If f : R ≃+* S is a ring isomorphism and I : Ideal R, then map f.symm I = comap f I.

theorem Ideal.mem_map_of_equiv {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 R} (y : S) :
y Ideal.map e I xI, e x = y
theorem Ideal.comap_map_of_surjective {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) :
def Ideal.relIsoOfSurjective {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 R // Ideal.comap f p }

Correspondence theorem

Equations
  • One or more equations did not get rendered due to their size.
def Ideal.orderEmbeddingOfSurjective {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) :

The map on ideals induced by a surjective map preserves inclusion.

Equations
  • One or more equations did not get rendered due to their size.
theorem Ideal.map_eq_top_or_isMaximal_of_surjective {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} (H : I.IsMaximal) :
Ideal.map f I = (Ideal.map f I).IsMaximal
theorem Ideal.comap_isMaximal_of_surjective {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
instance Ideal.comap_isMaximal_of_equiv {R : Type u} {S : Type v} [Ring R] [Ring S] {E : Type u_2} [EquivLike E R S] [RingEquivClass E R S] (e : E) {p : Ideal S} [p.IsMaximal] :
(Ideal.comap e p).IsMaximal

The pullback of a maximal ideal under a ring isomorphism is a maximal ideal.

Equations
  • =
theorem Ideal.comap_le_comap_iff_of_surjective {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) (J : Ideal S) :
def Ideal.relIsoOfBijective {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.Bijective f) :

Special case of the correspondence theorem for isomorphic rings

Equations
theorem Ideal.comap_le_iff_le_map {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.Bijective f) {I : Ideal R} {K : Ideal S} :
theorem Ideal.comap_map_of_bijective {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.Bijective f) {I : Ideal R} :
theorem Ideal.map.isMaximal {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.Bijective f) {I : Ideal R} (H : I.IsMaximal) :
(Ideal.map f I).IsMaximal
instance Ideal.map_isMaximal_of_equiv {R : Type u} {S : Type v} [Ring R] [Ring S] {E : Type u_2} [EquivLike E R S] [RingEquivClass E R S] (e : E) {p : Ideal R} [hp : p.IsMaximal] :
(Ideal.map e p).IsMaximal

A ring isomorphism sends a maximal ideal to a maximal ideal.

Equations
  • =
theorem Ideal.RingEquiv.bot_maximal_iff {R : Type u} {S : Type v} [Ring R] [Ring S] (e : R ≃+* S) :
.IsMaximal .IsMaximal
theorem Ideal.map_mul {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R) (J : Ideal R) :
Ideal.map f (I * J) = Ideal.map f I * Ideal.map f J
@[simp]
theorem Ideal.mapHom_apply {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R) :
def Ideal.mapHom {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) :

The pushforward Ideal.map as a monoid-with-zero homomorphism.

Equations
theorem Ideal.map_pow {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R) (n : ) :
Ideal.map f (I ^ n) = Ideal.map f I ^ n
theorem Ideal.comap_radical {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing 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
theorem Ideal.IsRadical.comap {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (hK : K.IsRadical) :
(Ideal.comap f K).IsRadical
theorem Ideal.map_radical_le {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {I : Ideal R} :
Ideal.map f I.radical (Ideal.map f I).radical
theorem Ideal.le_comap_mul {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} {L : Ideal S} :
theorem Ideal.le_comap_pow {R : Type u} {S : Type v} {F : Type u_1} [CommRing R] [CommRing S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (n : ) :
Ideal.comap f K ^ n Ideal.comap f (K ^ n)
def RingHom.ker {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) :

Kernel of a ring homomorphism as an ideal of the domain.

Equations
@[simp]
theorem RingHom.mem_ker {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} {r : R} :
r RingHom.ker f f r = 0

An element is in the kernel if and only if it maps to zero.

theorem RingHom.ker_eq {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) = f ⁻¹' {0}
theorem RingHom.ker_eq_comap_bot {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) :
theorem RingHom.comap_ker {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : S →+* R) (g : T →+* S) :
theorem RingHom.not_one_mem_ker {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Nontrivial S] (f : F) :
1RingHom.ker f

If the target is not the zero ring, then one is not in the kernel.

theorem RingHom.ker_ne_top {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rcf : RingHomClass F R S] [Nontrivial S] (f : F) :
theorem Pi.ker_ringHom {S : Type v} [Semiring S] {ι : Type u_3} {R : ιType u_4} [(i : ι) → Semiring (R i)] (φ : (i : ι) → S →+* R i) :
RingHom.ker (Pi.ringHom φ) = ⨅ (i : ι), RingHom.ker (φ i)
@[simp]
theorem RingHom.ker_rangeSRestrict {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R →+* S) :
RingHom.ker f.rangeSRestrict = RingHom.ker f
theorem RingHom.injective_iff_ker_eq_bot {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) :
theorem RingHom.ker_eq_bot_iff_eq_zero {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) :
RingHom.ker f = ∀ (x : R), f x = 0x = 0
@[simp]
theorem RingHom.ker_coe_equiv {R : Type u} {S : Type v} [Ring R] [Semiring S] (f : R ≃+* S) :
@[simp]
theorem RingHom.ker_equiv {R : Type u} {S : Type v} [Ring R] [Semiring S] {F' : Type u_2} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') :
theorem RingHom.sub_mem_ker_iff {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {x : R} {y : R} :
x - y RingHom.ker f f x = f y
@[simp]
theorem RingHom.ker_rangeRestrict {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
RingHom.ker f.rangeRestrict = RingHom.ker f
theorem RingHom.ker_isPrime {R : Type u} {S : Type v} {F : Type u_1} [Ring R] [Ring S] [IsDomain S] [FunLike F R S] [RingHomClass F R S] (f : F) :
(RingHom.ker f).IsPrime

The kernel of a homomorphism to a domain is a prime ideal.

theorem RingHom.ker_isMaximal_of_surjective {R : Type u_1} {K : Type u_2} {F : Type u_3} [Ring R] [Field K] [FunLike F R K] [RingHomClass F R K] (f : F) (hf : Function.Surjective f) :
(RingHom.ker f).IsMaximal

The kernel of a homomorphism to a field is a maximal ideal.

def Module.annihilator (R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] :

Module.annihilator R M is the ideal of all elements r : R such that r • M = 0.

Equations
theorem Module.mem_annihilator {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {r : R} :
r Module.annihilator R M ∀ (m : M), r m = 0
theorem LinearMap.annihilator_le_of_injective {R : Type u_1} {M : Type u_2} {M' : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Injective f) :
theorem LinearMap.annihilator_le_of_surjective {R : Type u_1} {M : Type u_2} {M' : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (f : M →ₗ[R] M') (hf : Function.Surjective f) :
theorem LinearEquiv.annihilator_eq {R : Type u_1} {M : Type u_2} {M' : Type u_3} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M'] [Module R M'] (e : M ≃ₗ[R] M') :
@[reducible, inline]
abbrev Submodule.annihilator {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M) :

N.annihilator is the ideal of all elements r : R such that r • N = 0.

Equations
theorem Submodule.annihilator_top {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] :
.annihilator = Module.annihilator R M
theorem Submodule.mem_annihilator {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {r : R} :
r N.annihilator nN, r n = 0
theorem Submodule.annihilator_bot {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] :
.annihilator =
theorem Submodule.annihilator_eq_top_iff {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} :
N.annihilator = N =
theorem Submodule.annihilator_mono {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {P : Submodule R M} (h : N P) :
P.annihilator N.annihilator
theorem Submodule.annihilator_iSup {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (ι : Sort w) (f : ιSubmodule R M) :
(⨆ (i : ι), f i).annihilator = ⨅ (i : ι), (f i).annihilator
theorem Submodule.mem_annihilator' {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {r : R} :
r N.annihilator N Submodule.comap (r LinearMap.id)
theorem Submodule.mem_annihilator_span {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (s : Set M) (r : R) :
r (Submodule.span R s).annihilator ∀ (n : s), r n = 0
theorem Submodule.mem_annihilator_span_singleton {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (g : M) (r : R) :
r (Submodule.span R {g}).annihilator r g = 0
@[simp]
theorem Submodule.annihilator_smul {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M) :
N.annihilator N =
theorem Ideal.map_eq_bot_iff_le_ker {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] {I : Ideal R} (f : F) :
theorem Ideal.ker_le_comap {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) :
instance Ideal.map_isPrime_of_equiv {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {F' : Type u_4} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') {I : Ideal R} [I.IsPrime] :
(Ideal.map f I).IsPrime

A ring isomorphism sends a prime ideal to a prime ideal.

Equations
  • =
theorem Ideal.comap_map_of_surjective' {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) :
theorem Ideal.map_sInf {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] {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
(∀ JA, RingHom.ker f J)Ideal.map f (sInf A) = sInf (Ideal.map f '' A)
theorem Ideal.map_isPrime_of_surjective {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} [H : I.IsPrime] (hk : RingHom.ker f I) :
(Ideal.map f I).IsPrime
theorem Ideal.map_eq_bot_iff_of_injective {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] {I : Ideal R} {f : F} (hf : Function.Injective f) :
theorem Ideal.map_eq_iff_sup_ker_eq_of_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {I : Ideal R} {J : Ideal R} (f : R →+* S) (hf : Function.Surjective f) :
theorem Ideal.map_radical_of_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {f : R →+* S} (hf : Function.Surjective f) {I : Ideal R} (h : RingHom.ker f I) :
Ideal.map f I.radical = (Ideal.map f I).radical
def RingHom.liftOfRightInverseAux {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : BA) (hf : Function.RightInverse f_inv f) (g : A →+* C) (hg : RingHom.ker f RingHom.ker g) :
B →+* C

Auxiliary definition used to define liftOfRightInverse

Equations
  • f.liftOfRightInverseAux f_inv hf g hg = { toFun := fun (b : B) => g (f_inv b), map_one' := , map_mul' := , map_zero' := , map_add' := }
@[simp]
theorem RingHom.liftOfRightInverseAux_comp_apply {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : BA) (hf : Function.RightInverse f_inv f) (g : A →+* C) (hg : RingHom.ker f RingHom.ker g) (a : A) :
(f.liftOfRightInverseAux f_inv hf g hg) (f a) = g a
def RingHom.liftOfRightInverse {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : BA) (hf : Function.RightInverse f_inv f) :
{ g : A →+* C // RingHom.ker f RingHom.ker g } (B →+* C)

liftOfRightInverse f hf g hg is the unique ring homomorphism φ

See RingHom.eq_liftOfRightInverse for the uniqueness lemma.

   A .
   |  \
 f |   \ g
   |    \
   v     \⌟
   B ----> C
      ∃!φ
Equations
  • One or more equations did not get rendered due to their size.
@[reducible, inline]
noncomputable abbrev RingHom.liftOfSurjective {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (hf : Function.Surjective f) :
{ g : A →+* C // RingHom.ker f RingHom.ker g } (B →+* C)

A non-computable version of RingHom.liftOfRightInverse for when no computable right inverse is available, that uses Function.surjInv.

Equations
theorem RingHom.liftOfRightInverse_comp_apply {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : BA) (hf : Function.RightInverse f_inv f) (g : { g : A →+* C // RingHom.ker f RingHom.ker g }) (x : A) :
((f.liftOfRightInverse f_inv hf) g) (f x) = g x
theorem RingHom.liftOfRightInverse_comp {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : BA) (hf : Function.RightInverse f_inv f) (g : { g : A →+* C // RingHom.ker f RingHom.ker g }) :
((f.liftOfRightInverse f_inv hf) g).comp f = g
theorem RingHom.eq_liftOfRightInverse {A : Type u_1} {B : Type u_2} {C : Type u_3} [Ring A] [Ring B] [Ring C] (f : A →+* B) (f_inv : BA) (hf : Function.RightInverse f_inv f) (g : A →+* C) (hg : RingHom.ker f RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) :
h = (f.liftOfRightInverse f_inv hf) g, hg
theorem AlgHom.coe_ker {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) :
theorem AlgHom.coe_ideal_map {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (I : Ideal A) :
Ideal.map f I = Ideal.map (↑f) I
theorem AlgHom.comap_ker {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) :
@[simp]
theorem Algebra.idealMap_apply_coe {R : Type u_1} [CommSemiring R] (S : Type u_2) [Semiring S] [Algebra R S] (I : Ideal R) (c : I) :
((Algebra.idealMap S I) c) = (algebraMap R S) c
def Algebra.idealMap {R : Type u_1} [CommSemiring R] (S : Type u_2) [Semiring S] [Algebra R S] (I : Ideal R) :
I →ₗ[R] (Ideal.map (algebraMap R S) I)

The induced linear map from I to the span of I in an R-algebra S.

Equations