Maps on modules and ideals

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] [rc : RingHomClass F R S] (f : F) (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] [rc : RingHomClass F R S] (f : F) (I : Ideal S) : ↑(Ideal.comap f I) = ⇑f ⁻¹' ↑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 : Ideal R} {J : Ideal R} (h : I ≤ J) : Ideal.map f I ≤ Ideal.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 ∈ 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] [rc : RingHomClass F R S] {f : F} {I : Ideal R} {K : Ideal S} : Ideal.map f I ≤ K ↔ I ≤ Ideal.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] [rc : RingHomClass F R S] {f : F} {K : Ideal 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] [rc : RingHomClass F R S] {f : F} {K : Ideal S} {L : Ideal S} (h : K ≤ L) : Ideal.comap f K ≤ Ideal.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] [rc : RingHomClass F R S] (f : F) {K : Ideal S} (hK : K ≠ ⊤) : Ideal.comap f 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] [rcg : RingHomClass G S R] (g : G) (I : Ideal R) (hf : Set.LeftInvOn ⇑g ⇑f ↑I) : Ideal.map f I ≤ Ideal.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] [rc : RingHomClass F R S] (f : F) {G : Type u_2} [FunLike G S R] (g : G) (I : Ideal S) (hf : Set.LeftInvOn (⇑g) (⇑f) (⇑f ⁻¹' ↑I)) : Ideal.comap f I ≤ Ideal.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] [rcg : RingHomClass G S R] (g : G) (I : Ideal R) (h : Function.LeftInverse ⇑g ⇑f) : Ideal.map f I ≤ Ideal.comap g I

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] [rc : RingHomClass F R S] (f : F) {G : Type u_2} [FunLike G S R] (g : G) (I : Ideal S) (h : Function.LeftInverse ⇑g ⇑f) : Ideal.comap f I ≤ Ideal.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] [rc : RingHomClass F R S] (f : F) {K : Ideal 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] [rc : RingHomClass F R S] (f : F) : Ideal.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] [rc : RingHomClass F R S] (f : F) : GaloisConnection (Ideal.map f) (Ideal.comap f)

@[simp]

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

@[simp]

theorem Ideal.map_id {R : Type u} [Semiring R] (I : Ideal R) : Ideal.map (RingHom.id R) 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) : 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_span {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (s : Set R) : Ideal.map f (Ideal.span s) = Ideal.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] [rc : RingHomClass F R S] {f : F} {I : Ideal R} {K : Ideal S} : I ≤ Ideal.comap f K → Ideal.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] [rc : RingHomClass F R S] {f : F} {I : Ideal R} {K : Ideal S} : Ideal.map f I ≤ K → I ≤ 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] [rc : RingHomClass F R S] {f : F} {I : Ideal R} : I ≤ Ideal.comap f (Ideal.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] [rc : RingHomClass F R S] {f : F} {K : Ideal S} : Ideal.map f (Ideal.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] [rc : RingHomClass F R S] {f : F} : Ideal.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] [rc : RingHomClass F R S] {f : F} {I : Ideal S} : Ideal.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] [rc : RingHomClass F R S] {f : F} : Ideal.map f ⊥ = ⊥

@[simp]

theorem Ideal.map_comap_map {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (I : Ideal R) : Ideal.map f (Ideal.comap f (Ideal.map f I)) = Ideal.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] [rc : RingHomClass F R S] (f : F) (K : Ideal S) : Ideal.comap f (Ideal.map f (Ideal.comap f K)) = Ideal.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] [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

theorem Ideal.comap_inf {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) (K : Ideal S) (L : Ideal S) : Ideal.comap f (K ⊓ L) = Ideal.comap f K ⊓ Ideal.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] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (K : ι → Ideal 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] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (K : ι → Ideal 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] [rc : RingHomClass F R S] (f : F) (s : Set (Ideal R)) : Ideal.map f (sSup s) = ⨆ I ∈ s, 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] [rc : RingHomClass F R S] (f : F) (s : Set (Ideal S)) : Ideal.comap f (sInf s) = ⨅ I ∈ s, 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] [rc : RingHomClass F R S] (f : F) (s : Set (Ideal S)) : Ideal.comap f (sInf s) = ⨅ I ∈ Ideal.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] [rc : RingHomClass F R S] (f : F) (K : Ideal S) [H : K.IsPrime] : (Ideal.comap f K).IsPrime

theorem Ideal.map_inf_le {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring 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

theorem Ideal.le_comap_sup {R : Type u} {S : Type v} {F : Type u_1} [Semiring R] [Semiring S] [FunLike F R S] [rc : RingHomClass F R S] (f : F) {K : Ideal S} {L : Ideal S} : Ideal.comap f K ⊔ Ideal.comap f L ≤ Ideal.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 (Ideal.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 (Ideal.map (algebraMap R S) I)

@[simp]

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

@[simp]

theorem Ideal.restrictScalars_mul {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] [Algebra R S] (I : Ideal S) (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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal S) : Ideal.map f (Ideal.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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) : GaloisInsertion (Ideal.map f) (Ideal.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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) : Function.Surjective (Ideal.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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) : Function.Injective (Ideal.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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal S) (J : Ideal S) : Ideal.map f (Ideal.comap f I ⊔ Ideal.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] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (hf : Function.Surjective ⇑f) (K : ι → Ideal 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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) (I : Ideal S) (J : Ideal S) : Ideal.map f (Ideal.comap f I ⊓ Ideal.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] [rc : RingHomClass F R S] (f : F) {ι : Sort u_3} (hf : Function.Surjective ⇑f) (K : ι → Ideal 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] [rc : RingHomClass F R S] (f : F) (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] [rc : RingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) {I : Ideal R} {y : S} : y ∈ Ideal.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] [rc : RingHomClass F R S] (f : F) {I : Ideal R} {K : Ideal S} (hf : Function.Surjective ⇑f) : Ideal.comap f K ≤ I → K ≤ 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] [rc : RingHomClass F R S] (f : F) {I : Ideal R} (hf : Function.Injective ⇑f) : Ideal.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] [rc : RingHomClass F R S] (f : F) (hf : Function.Injective ⇑f) : Ideal.comap 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.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) : Ideal.comap f (Ideal.map f I) = I ⊔ Ideal.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 // 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) : Ideal S ↪o Ideal R

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

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) : Ideal.comap f I ≤ Ideal.comap f J ↔ I ≤ J

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) : 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} [Ring R] [Ring S] [FunLike F R S] [RingHomClass F R S] (f : F) (hf : Function.Bijective ⇑f) {I : Ideal R} {K : Ideal S} : Ideal.comap f K ≤ I ↔ K ≤ Ideal.map f I

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

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) : (Ideal.mapHom f) I = Ideal.map f I

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) : Ideal R →*₀ Ideal S

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

Equations

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

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

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 : 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) : Ideal R

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

Equations

• RingHom.ker f = Ideal.comap f ⊥

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) : RingHom.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 (RingHom.ker f) = RingHom.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 ∉ RingHom.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) : RingHom.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) : 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) : Function.Injective ⇑f ↔ RingHom.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) : RingHom.ker f = ⊥ ↔ ∀ (x : R), f x = 0 → x = 0

@[simp]

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

@[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') : RingHom.ker 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.

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) : Ideal.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 ≤ Ideal.comap f K

theorem 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

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) → 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) : Ideal.map f I = ⊥ ↔ I = ⊥

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) : Ideal.map f I = Ideal.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) : 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 : B → A) (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

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

@[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 : 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 : B → A) (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

• 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 // 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

• 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 // 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 : B → A) (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 : B → A) (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) : RingHom.ker f = RingHom.ker ↑f

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

@[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

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