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) : Ideal S

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

Equations

• Ideal.map f I = Ideal.span (⇑f '' ↑I)

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) : Ideal R

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) : ↑(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) : comap (↑f) I = comap f I

theorem Ideal.map_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 R) : map (↑f) I = map f I

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 J : Ideal R} (h : I ≤ J) : map f I ≤ map f 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 ∈ 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 ∈ 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] : map f I ≤ K ↔ I ≤ comap f K

@[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 ∈ 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 L : Ideal S} [RingHomClass F R S] (h : K ≤ L) : comap f K ≤ comap f 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 ≠ ⊤) : comap f K ≠ ⊤

theorem Ideal.exists_ideal_comap_le_prime {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 : comap f I ≤ P) : ∃ Q ≥ I, Q.IsPrime ∧ comap f Q ≤ P

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) : map f I ≤ comap g 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)) : comap f I ≤ map g 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) : map f I ≤ comap g I

The Ideal version of Set.image_subset_preimage_of_inverse.

@[instance 100]

instance Ideal.instIsTwoSidedComap {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] : (comap f K).IsTwoSided

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) : comap f I ≤ map g I

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

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] : map f ⊤ = ⊤

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] : GaloisConnection (map f) (comap f)

@[simp]

theorem Ideal.comap_id {R : Type u} [Semiring R] (I : Ideal R) : comap (RingHom.id R) I = I

@[simp]

theorem Ideal.comap_idₐ {R : Type u_3} {S : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] (I : Ideal S) : comap (AlgHom.id R S) I = I

@[simp]

theorem Ideal.map_id {R : Type u} [Semiring R] (I : Ideal R) : map (RingHom.id R) I = I

@[simp]

theorem Ideal.map_idₐ {R : Type u_3} {S : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] (I : Ideal S) : map (AlgHom.id R S) I = I

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) : comap f (comap g I) = 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) : comap f (comap g I) = 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) : map g (map f I) = 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) : map g (map f I) = 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) : map f (span s) = span (⇑f '' s)

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 ≤ comap f K → 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] : map f I ≤ K → I ≤ 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] : I ≤ comap f (map f I)

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] : map f (comap f K) ≤ K

@[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] : comap f ⊤ = ⊤

@[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} : comap f I = ⊤ ↔ I = ⊤

@[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] : map f ⊥ = ⊥

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 : map f I ≠ ⊥) : 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] : map f (comap f (map f I)) = map f I

@[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] : comap f (map f (comap f K)) = comap f K

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 J : Ideal R) [RingHomClass F R S] : map f (I ⊔ J) = map f I ⊔ map f J

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 L : Ideal S) [RingHomClass F R S] : comap f (K ⊓ L) = comap f K ⊓ comap f L

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 : ι → Ideal R) : map f (iSup K) = ⨆ (i : ι), 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 : ι → Ideal S) : comap f (iInf K) = ⨅ (i : ι), 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)) : map f (sSup s) = ⨆ I ∈ s, 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)) : comap f (sInf s) = ⨅ I ∈ s, 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)) : comap f (sInf s) = ⨅ I ∈ 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] : (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 J : Ideal R} [RingHomClass F R S] : map f (I ⊓ J) ≤ map f I ⊓ map f J

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 L : Ideal S} [RingHomClass F R S] : comap f K ⊔ comap f L ≤ comap f (K ⊔ L)

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) : Submodule.restrictScalars R ((algebraMap R S) r • N) = r • Submodule.restrictScalars R N

theorem Ideal.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] (I : Ideal R) (N : Submodule S M) : Submodule.restrictScalars R (map (algebraMap R S) I • N) = I • Submodule.restrictScalars R N

@[simp]

theorem Ideal.smul_top_eq_map {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] [Algebra R S] (I : Ideal R) : I • ⊤ = Submodule.restrictScalars R (map (algebraMap R S) I)

@[simp]

theorem Ideal.coe_restrictScalars {R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [Module R S] [IsScalarTower R S S] (I : Ideal S) : ↑(Submodule.restrictScalars R I) = ↑I

@[simp]

theorem Ideal.restrictScalars_mul {R : Type u_4} {S : Type u_5} [Semiring R] [Semiring S] [Module R S] [IsScalarTower R S S] (I J : Ideal S) : Submodule.restrictScalars R (I * J) = Submodule.restrictScalars R I * Submodule.restrictScalars R J

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) : map f (comap f I) = I

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) : GaloisInsertion (map f) (comap f)

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

Equations

• Ideal.giMapComap f hf = GaloisInsertion.monotoneIntro ⋯ ⋯ ⋯ ⋯

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) : Function.Surjective (map 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) : Function.Injective (comap 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 J : Ideal S) : map f (comap f I ⊔ comap f J) = I ⊔ J

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 : ι → Ideal S) : map f (⨆ (i : ι), 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 J : Ideal S) : map f (comap f I ⊓ comap f J) = I ⊓ J

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 : ι → Ideal S) : map f (⨅ (i : ι), 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 ∈ 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 ∈ map f I ↔ ∃ x ∈ I, 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) : comap f K ≤ I → K ≤ map f I

theorem Ideal.map_comap_eq_self_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 S) : map e (comap e I) = 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) : map f I = Submodule.map f.toSemilinearMap I

@[instance 100]

instance Ideal.instIsTwoSidedMapRingHomOfRingHomSurjective {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R →+* S) [RingHomSurjective f] (I : Ideal R) [I.IsTwoSided] : (map f I).IsTwoSided

theorem Ideal.IsMaximal.comap_piEvalRingHom {ι : Type u_4} {R : ι → Type u_5} [(i : ι) → Semiring (R i)] {i : ι} {I : Ideal (R i)} (h : I.IsMaximal) : (comap (Pi.evalRingHom R i) I).IsMaximal

theorem Ideal.comap_le_comap_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 J : Ideal S) : comap f I ≤ comap f J ↔ I ≤ J

def Ideal.orderEmbeddingOfSurjective {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) : Ideal S ↪o Ideal R

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

Equations

• Ideal.orderEmbeddingOfSurjective f hf = { toFun := Ideal.comap f, inj' := ⋯, map_rel_iff' := ⋯ }

theorem Ideal.map_eq_top_or_isMaximal_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} (H : I.IsMaximal) : map f I = ⊤ ∨ (map f I).IsMaximal

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) : comap f ⊥ ≤ I

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) : comap f ⊥ = ⊥

@[simp]

theorem Ideal.map_of_equiv {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} (f : R ≃+* S) : map (↑f.symm) (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) : comap (↑f) (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) : map (↑f) I = 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) : comap f.symm I = 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) : map f.symm I = comap f I

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

@[simp]

theorem Ideal.symm_apply_mem_of_equiv_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} {f : R ≃+* S} {y : S} : f.symm y ∈ I ↔ y ∈ map f I

@[simp]

theorem Ideal.apply_mem_of_equiv_iff {R : Type u} {S : Type v} [Semiring R] [Semiring S] {I : Ideal R} {f : R ≃+* S} {x : R} : f x ∈ map f I ↔ x ∈ 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 ∈ map e I ↔ ∃ x ∈ I, e x = y

def Ideal.relIsoOfBijective {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) : Ideal S ≃o Ideal R

Special case of the correspondence theorem for isomorphic rings

Equations

• Ideal.relIsoOfBijective f hf = { toFun := Ideal.comap f, invFun := Ideal.map f, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }

theorem Ideal.comap_le_iff_le_map {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} : comap f K ≤ I ↔ K ≤ map f I

theorem Ideal.comap_map_of_bijective {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} : comap f (map f I) = I

theorem Ideal.isMaximal_map_iff_of_bijective {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} : (map f I).IsMaximal ↔ I.IsMaximal

theorem Ideal.isMaximal_comap_iff_of_bijective {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} : (comap f K).IsMaximal ↔ K.IsMaximal

theorem Ideal.IsMaximal.map_bijective {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} : I.IsMaximal → (map f I).IsMaximal

Alias of the reverse direction of Ideal.isMaximal_map_iff_of_bijective.

theorem Ideal.IsMaximal.comap_bijective {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 → (comap f K).IsMaximal

Alias of the reverse direction of Ideal.isMaximal_comap_iff_of_bijective.

instance Ideal.map_isMaximal_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) {p : Ideal R} [hp : p.IsMaximal] : (map e p).IsMaximal

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

instance Ideal.comap_isMaximal_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) {p : Ideal S} [hp : p.IsMaximal] : (comap e p).IsMaximal

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

theorem Ideal.isMaximal_iff_of_bijective {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) : ⊥.IsMaximal ↔ ⊥.IsMaximal

@[deprecated Ideal.IsMaximal.map_bijective (since := "2024-12-07")]

theorem Ideal.map.isMaximal {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} : I.IsMaximal → (map f I).IsMaximal

Alias of the reverse direction of Ideal.isMaximal_map_iff_of_bijective.

---

Alias of the reverse direction of Ideal.isMaximal_map_iff_of_bijective.

@[deprecated Ideal.IsMaximal.comap_bijective (since := "2024-12-07")]

theorem Ideal.comap.isMaximal {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 → (comap f K).IsMaximal

Alias of the reverse direction of Ideal.isMaximal_comap_iff_of_bijective.

---

Alias of the reverse direction of Ideal.isMaximal_comap_iff_of_bijective.

@[deprecated Ideal.isMaximal_iff_of_bijective (since := "2024-12-07")]

theorem Ideal.RingEquiv.bot_maximal_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] (hf : Function.Bijective ⇑f) : ⊥.IsMaximal ↔ ⊥.IsMaximal

Alias of Ideal.isMaximal_iff_of_bijective.

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) : comap f (map f I) = I ⊔ comap f ⊥

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 // comap f ⊥ ≤ p }

Correspondence theorem

Equations

• One or more equations did not get rendered due to their size.

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] : (comap f K).IsMaximal

theorem Ideal.map_mul {S : Type v} {F : Type u_1} [CommSemiring S] {R : Type u_2} [Semiring R] [FunLike F R S] [RingHomClass F R S] (f : F) (I J : Ideal R) : map f (I * J) = map f I * map f J

def Ideal.mapHom {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) : Ideal R →+* Ideal S

The pushforward Ideal.map as a (semi)ring homomorphism.

Equations

• Ideal.mapHom f = { toFun := Ideal.map f, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Ideal.mapHom_apply {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) (I : Ideal R) : (mapHom f) I = map f I

theorem Ideal.map_pow {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) (I : Ideal R) (n : ℕ) : map f (I ^ n) = map f I ^ n

theorem Ideal.comap_radical {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) : comap f K.radical = (comap f K).radical

theorem Ideal.IsRadical.comap {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

theorem Ideal.map_radical_le {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) {I : Ideal R} : map f I.radical ≤ (map f I).radical

theorem Ideal.le_comap_mul {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} : comap f K * comap f L ≤ comap f (K * L)

theorem Ideal.le_comap_pow {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) : comap f K ^ n ≤ comap f (K ^ n)

theorem Ideal.disjoint_map_primeCompl_iff_comap_le {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) ↔ comap f I ≤ p

theorem Ideal.comap_map_eq_self_iff_of_isPrime {R : Type u} [CommSemiring R] {S : Type u_2} [CommSemiring S] {f : R →+* S} (p : Ideal R) [p.IsPrime] : comap f (map f p) = p ↔ ∃ (q : Ideal S), q.IsPrime ∧ comap f q = p

For a prime ideal p of R, p extended to S and restricted back to R is p if and only if p is the restriction of a prime in S.

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) : Ideal R

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

Equations

• RingHom.ker f = Ideal.comap f ⊥

@[instance 100]

instance RingHom.instIsTwoSidedKer {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) : (ker f).IsTwoSided

@[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 ∈ 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) : ↑(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) : ker f = Ideal.comap 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) : Ideal.comap g (ker f) = ker (f.comp g)

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) : 1 ∉ 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) : ker f ≠ ⊤

theorem Pi.ker_ringHom {S : Type v} [Semiring S] {ι : Type u_3} {R : ι → Type 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) : ker f.rangeSRestrict = ker f

@[simp]

theorem RingHom.ker_coe_equiv {R : Type u} {S : Type v} [Semiring R] [Semiring S] (f : R ≃+* S) : ker ↑f = ⊥

theorem RingHom.ker_coe_toRingHom {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) : ker ↑f = ker f

@[simp]

theorem RingHom.ker_equiv {R : Type u} {S : Type v} [Semiring R] [Semiring S] {F' : Type u_3} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') : ker f = ⊥

theorem RingHom.ker_equiv_comp {R : Type u} {S : Type v} {T : Type w} [Semiring R] [Semiring S] [Semiring T] (f : R →+* S) (e : S ≃+* T) : ker (e.toRingHom.comp f) = 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) : Function.Injective ⇑f ↔ ker 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) : ker f = ⊥ ↔ ∀ (x : R), f x = 0 → x = 0

theorem RingHom.ker_comp_of_injective {R : Type u} {S : Type v} {T : Type w} [Ring R] [Semiring S] [Semiring T] (g : T →+* R) {f : R →+* S} (hf : Function.Injective ⇑f) : ker (f.comp g) = ker g

@[simp]

theorem AlgHom.ker_coe_equiv {R : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (e : A ≃ₐ[R] B) : RingHom.ker ↑e = ⊥

Synonym for RingHom.ker_coe_equiv, but given an algebra equivalence.

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 y : R} : x - y ∈ ker f ↔ f x = f y

@[simp]

theorem RingHom.ker_rangeRestrict {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : ker f.rangeRestrict = ker f

theorem RingHom.ker_isPrime {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [IsDomain S] [FunLike F R S] [RingHomClass F R S] (f : F) : (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] [DivisionRing K] [FunLike F R K] [RingHomClass F R K] (f : F) (hf : Function.Surjective ⇑f) : (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] : Ideal R

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

Equations

• Module.annihilator R M = RingHom.ker (Module.toAddMonoidEnd R M)

theorem Module.mem_annihilator {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {r : R} : r ∈ annihilator R M ↔ ∀ (m : M), r • m = 0

@[instance 100]

instance instIsTwoSidedAnnihilator {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] : (Module.annihilator R M).IsTwoSided

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) : Module.annihilator R M' ≤ Module.annihilator R M

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) : Module.annihilator R M ≤ Module.annihilator R M'

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') : Module.annihilator R M = Module.annihilator R M'

theorem Module.comap_annihilator {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) (annihilator R M) = annihilator R₀ M

theorem Module.annihilator_eq_bot {R : Type u_4} {M : Type u_5} [Ring R] [AddCommGroup M] [Module R M] : annihilator R M = ⊥ ↔ FaithfulSMul R M

theorem Module.annihilator_eq_top_iff {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] : annihilator R M = ⊤ ↔ Subsingleton 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) : Ideal R

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

Equations

• N.annihilator = Module.annihilator R ↥N

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 ↔ ∀ n ∈ N, 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 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 : ι → Submodule R M) : (⨆ (i : ι), f i).annihilator = ⨅ (i : ι), (f i).annihilator

theorem Submodule.le_annihilator_iff {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M} {I : Ideal R} : I ≤ N.annihilator ↔ I • N = ⊥

@[simp]

theorem Submodule.annihilator_smul {R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [Module R M] (N : Submodule R M) : N.annihilator • N = ⊥

@[simp]

theorem Submodule.annihilator_mul {R : Type u_1} [Semiring R] (I : Ideal R) : annihilator I * I = ⊥

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 ≤ 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 ∈ (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 ∈ (span R {g}).annihilator ↔ r • g = 0

theorem Submodule.annihilator_span {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (s : Set M) : (span R s).annihilator = ⨅ (g : ↑s), LinearMap.ker (LinearMap.toSpanSingleton R M ↑g)

theorem Submodule.annihilator_span_singleton {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] (g : M) : (span R {g}).annihilator = LinearMap.ker (LinearMap.toSpanSingleton R M g)

@[simp]

theorem Submodule.mul_annihilator {R : Type u_1} [CommSemiring R] (I : Ideal R) : I * annihilator I = ⊥

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) : map f I = ⊥ ↔ I ≤ RingHom.ker 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) : RingHom.ker f ≤ comap f K

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] : (map f I).IsPrime

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

theorem Ideal.map_eq_bot_iff_of_injective {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} (hf : Function.Injective ⇑f) : map f I = ⊥ ↔ I = ⊥

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) : comap f (map f I) = I ⊔ RingHom.ker f

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) : (∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (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) : (map f I).IsPrime

theorem Ideal.map_ne_bot_of_ne_bot {R : Type u_1} [CommRing R] {S : Type u_4} [Ring S] [Nontrivial S] [Algebra R S] [NoZeroSMulDivisors R S] {I : Ideal R} (h : I ≠ ⊥) : map (algebraMap R S) I ≠ ⊥

theorem Ideal.map_eq_iff_sup_ker_eq_of_surjective {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] {I J : Ideal R} (f : R →+* S) (hf : Function.Surjective ⇑f) : map f I = map f J ↔ I ⊔ RingHom.ker f = J ⊔ RingHom.ker 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) : map f I.radical = (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 : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : ker f ≤ 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 : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : ker f ≤ 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 : B → A) (hf : Function.RightInverse f_inv ⇑f) : { g : A →+* C // ker f ≤ ker g } ≃ (B →+* C)

liftOfRightInverse f hf g hg is the unique ring homomorphism φ

• such that φ.comp f = g (RingHom.liftOfRightInverse_comp),
• where f : A →+* B has a right_inverse f_inv (hf),
• and g : B →+* C satisfies hg : f.ker ≤ g.ker.

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 // ker f ≤ ker g } ≃ (B →+* C)

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

Equations

• f.liftOfSurjective hf = f.liftOfRightInverse (Function.surjInv hf) ⋯

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 : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : { g : A →+* C // ker f ≤ 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 : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : { g : A →+* C // ker f ≤ 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 : B → A) (hf : Function.RightInverse f_inv ⇑f) (g : A →+* C) (hg : ker f ≤ ker g) (h : B →+* C) (hh : h.comp f = g) : h = (f.liftOfRightInverse f_inv hf) ⟨g, hg⟩

theorem AlgHom.ker_coe {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) : RingHom.ker f = RingHom.ker ↑f

@[deprecated AlgHom.ker_coe (since := "2025-02-24")]

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) : RingHom.ker f = RingHom.ker ↑f

Alias of AlgHom.ker_coe.

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) : Ideal.comap g (RingHom.ker f) = RingHom.ker (f.comp g)

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

• Algebra.idealMap S I = (Algebra.linearMap R S).restrict ⋯

@[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) : ↑((idealMap S I) c) = (algebraMap R S) ↑c

@[simp]

theorem FaithfulSMul.ker_algebraMap_eq_bot (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] : RingHom.ker (algebraMap R A) = ⊥

@[deprecated FaithfulSMul.ker_algebraMap_eq_bot (since := "2025-01-31")]

theorem NoZeroSMulDivisors.iff_ker_algebraMap_eq_bot (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] : RingHom.ker (algebraMap R A) = ⊥

Alias of FaithfulSMul.ker_algebraMap_eq_bot.

@[deprecated FaithfulSMul.ker_algebraMap_eq_bot (since := "2025-01-31")]

theorem NoZeroSMulDivisors.of_ker_algebraMap_eq_bot (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] : RingHom.ker (algebraMap R A) = ⊥

Alias of FaithfulSMul.ker_algebraMap_eq_bot.

@[deprecated FaithfulSMul.ker_algebraMap_eq_bot (since := "2025-01-31")]

theorem NoZeroSMulDivisors.ker_algebraMap_eq_bot (R : Type u_1) (A : Type u_2) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] : RingHom.ker (algebraMap R A) = ⊥

Alias of FaithfulSMul.ker_algebraMap_eq_bot.