Homomorphisms of R-algebras

This file defines bundled homomorphisms of R-algebras.

Main definitions

• AlgHom R A B: the type of R-algebra morphisms from A to B.
• Algebra.ofId R A : R →ₐ[R] A: the canonical map from R to A, as an AlgHom.

Notation

• A →ₐ[R] B : R-algebra homomorphism from A to B.

structure AlgHom (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] extends A →+* B : Type (max v w)

Defining the homomorphism in the category R-Alg, denoted A →ₐ[R] B.

• toFun : A → B
• map_one' : (↑↑self.toRingHom).toFun 1 = 1
• map_mul' (x y : A) : (↑↑self.toRingHom).toFun (x * y) = (↑↑self.toRingHom).toFun x * (↑↑self.toRingHom).toFun y
• map_zero' : (↑↑self.toRingHom).toFun 0 = 0
• map_add' (x y : A) : (↑↑self.toRingHom).toFun (x + y) = (↑↑self.toRingHom).toFun x + (↑↑self.toRingHom).toFun y
• commutes' (r : R) : (↑↑self.toRingHom).toFun ((algebraMap R A) r) = (algebraMap R B) r

def «term_→ₐ_» : Lean.TrailingParserDescr

Defining the homomorphism in the category R-Alg, denoted A →ₐ[R] B.

Equations

• «term_→ₐ_» = Lean.ParserDescr.trailingNode `«term_→ₐ_» 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " →ₐ ") (Lean.ParserDescr.cat `term 25))

def «term_→ₐ[_]_» : Lean.TrailingParserDescr

Defining the homomorphism in the category R-Alg, denoted A →ₐ[R] B.

Equations

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

class AlgHomClass (F : Type u_1) (R : outParam (Type u_2)) (A : outParam (Type u_3)) (B : outParam (Type u_4)) [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [FunLike F A B] extends RingHomClass F A B : Prop

AlgHomClass F R A B asserts F is a type of bundled algebra homomorphisms from A to B.

• map_mul (f : F) (x y : A) : f (x * y) = f x * f y
• map_one (f : F) : f 1 = 1
• map_add (f : F) (x y : A) : f (x + y) = f x + f y
• map_zero (f : F) : f 0 = 0
• commutes (f : F) (r : R) : f ((algebraMap R A) r) = (algebraMap R B) r

@[instance 100]

instance AlgHomClass.linearMapClass {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [FunLike F A B] [AlgHomClass F R A B] : LinearMapClass F R A B

def AlgHom.ofClass {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 : Type u_5} [FunLike F A B] [AlgHomClass F R A B] (f : F) : A →ₐ[R] B

Turn an element of a type F satisfying AlgHomClass F α β into an actual AlgHom. This is declared as the default coercion from F to α →+* β.

Equations

• ↑f = { toFun := ⇑f, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[deprecated AlgHom.ofClass (since := "2026-09-07")]

def AlgHomClass.toAlgHom {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 : Type u_5} [FunLike F A B] [AlgHomClass F R A B] (f : F) : A →ₐ[R] B

Alias of AlgHom.ofClass.

---

Turn an element of a type F satisfying AlgHomClass F α β into an actual AlgHom. This is declared as the default coercion from F to α →+* β.

Equations

• @AlgHomClass.toAlgHom = @AlgHom.ofClass

@[instance_reducible, macro_inline]

instance AlgHom.funLike {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : FunLike (A →ₐ[R] B) A B

Equations

• AlgHom.funLike = { coe := fun (f : A →ₐ[R] B) => (↑↑f.toRingHom).toFun, coe_injective := ⋯ }

instance AlgHom.algHomClass {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : AlgHomClass (A →ₐ[R] B) R A B

@[simp]

theorem AlgHomClass.linearMapOfClass_ofClass {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [FunLike F A B] [AlgHomClass F R A B] (f : F) : ↑↑f = ↑f

@[deprecated AlgHomClass.linearMapOfClass_ofClass (since := "2026-09-08")]

theorem AlgHomClass.toLinearMap_toAlgHom {R : Type u_1} {A : Type u_2} {B : Type u_3} {F : Type u_4} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [FunLike F A B] [AlgHomClass F R A B] (f : F) : ↑↑f = ↑f

Alias of AlgHomClass.linearMapOfClass_ofClass.

def AlgHom.Simps.apply {R : Type u} {α : Type v} {β : Type w} [CommSemiring R] [Semiring α] [Semiring β] [Algebra R α] [Algebra R β] (f : α →ₐ[R] β) : α → β

See Note [custom simps projection]

Equations

• AlgHom.Simps.apply f = ⇑f

@[simp]

theorem AlgHom.coe_ofClass {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {F : Type u_1} [FunLike F A B] [AlgHomClass F R A B] (f : F) : ⇑↑f = ⇑f

@[deprecated AlgHom.coe_ofClass (since := "2026-09-08")]

theorem AlgHom.coe_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {F : Type u_1} [FunLike F A B] [AlgHomClass F R A B] (f : F) : ⇑↑f = ⇑f

Alias of AlgHom.coe_ofClass.

@[simp]

theorem AlgHom.toFun_eq_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : (↑↑f.toRingHom).toFun = ⇑f

def AlgHom.toMonoidHom' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : A →* B

Turn an algebra homomorphism into the corresponding multiplicative monoid homomorphism.

Equations

• ↑f = ↑↑f

@[instance_reducible]

instance AlgHom.coeOutMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : CoeOut (A →ₐ[R] B) (A →* B)

Equations

• AlgHom.coeOutMonoidHom = { coe := AlgHom.toMonoidHom' }

def AlgHom.toAddMonoidHom' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : A →+ B

Turn an algebra homomorphism into the corresponding additive monoid homomorphism.

Equations

• ↑f = ↑↑f

@[instance_reducible]

instance AlgHom.coeOutAddMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : CoeOut (A →ₐ[R] B) (A →+ B)

Equations

• AlgHom.coeOutAddMonoidHom = { coe := AlgHom.toAddMonoidHom' }

@[simp]

theorem AlgHom.coe_mk {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →+* B} (h : ∀ (r : R), (↑↑f).toFun ((algebraMap R A) r) = (algebraMap R B) r) : ⇑{ toRingHom := f, commutes' := h } = ⇑f

theorem AlgHom.coe_mks {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A → B} (h₁ : f 1 = 1) (h₂ : ∀ (x y : A), { toFun := f, map_one' := h₁ }.toFun (x * y) = { toFun := f, map_one' := h₁ }.toFun x * { toFun := f, map_one' := h₁ }.toFun y) (h₃ : (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0) (h₄ : ∀ (x y : A), (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) = (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun x + (↑{ toFun := f, map_one' := h₁, map_mul' := h₂ }).toFun y) (h₅ : ∀ (r : R), (↑↑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) = (algebraMap R B) r) : ⇑{ toFun := f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ } = f

@[simp]

theorem AlgHom.toRingHom_mk {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →+* B} (h : ∀ (r : R), (↑↑f).toFun ((algebraMap R A) r) = (algebraMap R B) r) : ↑{ toRingHom := f, commutes' := h } = f

@[deprecated AlgHom.toRingHom_mk (since := "2026-05-05")]

theorem AlgHom.coe_ringHom_mk {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →+* B} (h : ∀ (r : R), (↑↑f).toFun ((algebraMap R A) r) = (algebraMap R B) r) : ↑{ toRingHom := f, commutes' := h } = f

Alias of AlgHom.toRingHom_mk.

@[simp]

theorem AlgHom.toRingHom_eq_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : f.toRingHom = ↑f

@[simp]

theorem AlgHom.coe_toRingHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ⇑↑f = ⇑f

@[simp]

theorem AlgHom.coe_toMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ⇑↑f = ⇑f

@[simp]

theorem AlgHom.coe_toAddMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ⇑↑f = ⇑f

@[simp]

theorem AlgHom.toRingHom_toMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ↑↑f = ↑f

@[simp]

theorem AlgHom.toRingHom_toAddMonoidHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : ↑↑f = ↑f

theorem AlgHom.coe_fn_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective DFunLike.coe

theorem AlgHom.coe_fn_inj {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ φ₂ : A →ₐ[R] B} : ⇑φ₁ = ⇑φ₂ ↔ φ₁ = φ₂

theorem AlgHom.toRingHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective RingHomClass.toRingHom

@[deprecated AlgHom.toRingHom_injective (since := "2026-05-05")]

theorem AlgHom.coe_ringHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective RingHomClass.toRingHom

Alias of AlgHom.toRingHom_injective.

theorem AlgHom.toMonoidHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective MonoidHom.ofClass

@[deprecated AlgHom.toMonoidHom_injective (since := "2026-09-15")]

theorem AlgHom.coe_monoidHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective MonoidHom.ofClass

Alias of AlgHom.toMonoidHom_injective.

theorem AlgHom.toAddMonoidHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective AddMonoidHom.ofClass

@[deprecated AlgHom.toAddMonoidHom_injective (since := "2026-09-15")]

theorem AlgHom.coe_addMonoidHom_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective AddMonoidHom.ofClass

Alias of AlgHom.toAddMonoidHom_injective.

theorem AlgHom.congr_fun {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ φ₂ : A →ₐ[R] B} (H : φ₁ = φ₂) (x : A) : φ₁ x = φ₂ x

theorem AlgHom.congr_arg {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {x y : A} (h : x = y) : φ x = φ y

theorem AlgHom.ext {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ φ₂ : A →ₐ[R] B} (H : ∀ (x : A), φ₁ x = φ₂ x) : φ₁ = φ₂

theorem AlgHom.ext_iff {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {φ₁ φ₂ : A →ₐ[R] B} : φ₁ = φ₂ ↔ ∀ (x : A), φ₁ x = φ₂ x

@[simp]

theorem AlgHom.mk_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] {f : A →ₐ[R] B} (h₁ : f 1 = 1) (h₂ : ∀ (x y : A), { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) = { toFun := ⇑f, map_one' := h₁ }.toFun x * { toFun := ⇑f, map_one' := h₁ }.toFun y) (h₃ : (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun 0 = 0) (h₄ : ∀ (x y : A), (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun (x + y) = (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun x + (↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂ }).toFun y) (h₅ : ∀ (r : R), (↑↑{ toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄ }).toFun ((algebraMap R A) r) = (algebraMap R B) r) : { toFun := ⇑f, map_one' := h₁, map_mul' := h₂, map_zero' := h₃, map_add' := h₄, commutes' := h₅ } = f

@[simp]

theorem AlgHom.addHomMk_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : { toFun := ⇑f, map_add' := ⋯ } = ↑f

@[simp]

theorem AlgHom.commutes {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (r : R) : φ ((algebraMap R A) r) = (algebraMap R B) r

theorem AlgHom.comp_algebraMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : (↑φ).comp (algebraMap R A) = algebraMap R B

def AlgHom.mk' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →+* B) (h : ∀ (c : R) (x : A), f (c • x) = c • f x) : A →ₐ[R] B

If a RingHom is R-linear, then it is an AlgHom.

Equations

• AlgHom.mk' f h = { toFun := ⇑f, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem AlgHom.coe_mk' {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →+* B) (h : ∀ (c : R) (x : A), f (c • x) = c • f x) : ⇑(mk' f h) = ⇑f

def AlgHom.id (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : A →ₐ[R] A

Identity map as an AlgHom.

Equations

• AlgHom.id R A = { toRingHom := RingHom.id A, commutes' := ⋯ }

@[simp]

theorem AlgHom.coe_id (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : ⇑(AlgHom.id R A) = id

@[simp]

theorem AlgHom.id_toRingHom (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : ↑(AlgHom.id R A) = RingHom.id A

theorem AlgHom.id_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (p : A) : (AlgHom.id R A) p = p

def AlgHom.comp {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : A →ₐ[R] C

If φ₁ and φ₂ are R-algebra homomorphisms with the domain of φ₁ equal to the codomain of φ₂, then φ₁.comp φ₂ is the algebra homomorphism x ↦ φ₁ (φ₂ x).

Equations

• φ₁.comp φ₂ = { toRingHom := φ₁.comp ↑φ₂, commutes' := ⋯ }

@[simp]

theorem AlgHom.coe_comp {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ⇑(φ₁.comp φ₂) = ⇑φ₁ ∘ ⇑φ₂

theorem AlgHom.comp_apply {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) (p : A) : (φ₁.comp φ₂) p = φ₁ (φ₂ p)

theorem AlgHom.comp_toRingHom {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (φ₁ : B →ₐ[R] C) (φ₂ : A →ₐ[R] B) : ↑(φ₁.comp φ₂) = (↑φ₁).comp ↑φ₂

@[simp]

theorem AlgHom.comp_id {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : φ.comp (AlgHom.id R A) = φ

@[simp]

theorem AlgHom.id_comp {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : (AlgHom.id R B).comp φ = φ

theorem AlgHom.comp_assoc {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Semiring D] [Algebra R A] [Algebra R B] [Algebra R C] [Algebra R D] (φ₁ : C →ₐ[R] D) (φ₂ : B →ₐ[R] C) (φ₃ : A →ₐ[R] B) : (φ₁.comp φ₂).comp φ₃ = φ₁.comp (φ₂.comp φ₃)

instance AlgHom.instRingHomCompTripleComp {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] {φ₁ : B →ₐ[R] C} {φ₂ : A →ₐ[R] B} : RingHomCompTriple φ₂.toRingHom φ₁.toRingHom (φ₁.comp φ₂).toRingHom

def AlgHom.toLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : A →ₗ[R] B

R-Alg ⥤ R-Mod

Equations

• φ.toLinearMap = { toFun := ⇑φ, map_add' := ⋯, map_smul' := ⋯ }

theorem AlgHom.toLinearMap_eq_coe {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : f.toLinearMap = ↑f

@[simp]

theorem AlgHom.toLinearMap_apply {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (p : A) : φ.toLinearMap p = φ p

@[simp]

theorem AlgHom.coe_toLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) : ⇑φ.toLinearMap = ⇑φ

theorem AlgHom.toLinearMap_injective {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] : Function.Injective toLinearMap

@[simp]

theorem AlgHom.comp_toLinearMap {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] (f : A →ₐ[R] B) (g : B →ₐ[R] C) : (g.comp f).toLinearMap = g.toLinearMap ∘ₗ f.toLinearMap

@[simp]

theorem AlgHom.toLinearMap_id {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : (AlgHom.id R A).toLinearMap = LinearMap.id

@[simp]

theorem AlgHom.linearMapMk_toAddHom {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) : { toAddHom := ↑f, map_smul' := ⋯ } = f.toLinearMap

def AlgHom.ofLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) : A →ₐ[R] B

Promote a LinearMap to an AlgHom by supplying proofs about the behavior on 1 and *.

Equations

• AlgHom.ofLinearMap f map_one map_mul = { toFun := ⇑f, map_one' := map_one, map_mul' := map_mul, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem AlgHom.ofLinearMap_apply {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) (a : A) : (ofLinearMap f map_one map_mul) a = f a

@[simp]

theorem AlgHom.ofLinearMap_toLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) (map_one : φ.toLinearMap 1 = 1) (map_mul : ∀ (x y : A), φ.toLinearMap (x * y) = φ.toLinearMap x * φ.toLinearMap y) : ofLinearMap φ.toLinearMap map_one map_mul = φ

@[simp]

theorem AlgHom.toLinearMap_ofLinearMap {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₗ[R] B) (map_one : f 1 = 1) (map_mul : ∀ (x y : A), f (x * y) = f x * f y) : (ofLinearMap f map_one map_mul).toLinearMap = f

@[simp]

theorem AlgHom.ofLinearMap_id {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (map_one : LinearMap.id 1 = 1) (map_mul : ∀ (x y : A), LinearMap.id (x * y) = LinearMap.id x * LinearMap.id y) : ofLinearMap LinearMap.id map_one map_mul = AlgHom.id R A

theorem AlgHom.map_smul_of_tower {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (φ : A →ₐ[R] B) {R' : Type u_1} [SMul R' A] [SMul R' B] [LinearMap.CompatibleSMul A B R' R] (r : R') (x : A) : φ (r • x) = r • φ x

@[instance_reducible]

instance AlgHom.End {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : Monoid (A →ₐ[R] A)

Equations

• AlgHom.End = { one := AlgHom.id R A, mul := AlgHom.comp, mul_assoc := ⋯, one_mul := ⋯, mul_one := ⋯, npow_zero := ⋯, npow_succ := ⋯ }

theorem AlgHom.End_toOne_one {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : 1 = AlgHom.id R A

theorem AlgHom.End_toMul_mul {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ₁ φ₂ : A →ₐ[R] A) : φ₁ * φ₂ = φ₁.comp φ₂

@[simp]

theorem AlgHom.one_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (x : A) : 1 x = x

@[simp]

theorem AlgHom.mul_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ ψ : A →ₐ[R] A) (x : A) : (φ * ψ) x = φ (ψ x)

@[simp]

theorem AlgHom.coe_pow {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ : A →ₐ[R] A) (n : ℕ) : ⇑(φ ^ n) = (⇑φ)^[n]

theorem AlgHom.algebraMap_eq_apply {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) {y : R} {x : A} (h : (algebraMap R A) y = x) : (algebraMap R B) y = f x

theorem AlgHom.cancel_right {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] {g₁ g₂ : B →ₐ[R] C} {f : A →ₐ[R] B} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

theorem AlgHom.cancel_left {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [CommSemiring R] [Semiring A] [Semiring B] [Semiring C] [Algebra R A] [Algebra R B] [Algebra R C] {g₁ g₂ : A →ₐ[R] B} {f : B →ₐ[R] C} (hf : Function.Injective ⇑f) : f.comp g₁ = f.comp g₂ ↔ g₁ = g₂

def AlgHom.toEnd {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] : (A →ₐ[R] A) →* Module.End R A

AlgHom.toLinearMap as a MonoidHom.

Equations

• AlgHom.toEnd = { toFun := AlgHom.toLinearMap, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem AlgHom.toEnd_apply {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (φ : A →ₐ[R] A) : toEnd φ = φ.toLinearMap

def IsScalarTower.toAlgHom (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] : S →ₐ[R] A

In a tower, the canonical map from the middle element to the top element is an algebra homomorphism over the bottom element.

Equations

• IsScalarTower.toAlgHom R S A = { toRingHom := algebraMap S A, commutes' := ⋯ }

theorem IsScalarTower.toAlgHom_apply (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (y : S) : (toAlgHom R S A) y = (algebraMap S A) y

@[simp]

theorem IsScalarTower.coe_toAlgHom (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] : ↑(toAlgHom R S A) = algebraMap S A

@[simp]

theorem IsScalarTower.coe_toAlgHom' (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] : ⇑(toAlgHom R S A) = ⇑(algebraMap S A)

def Algebra.algHom (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] : S →ₐ[R] A

The algebra morphism underlying algebraMap.

Equations

• Algebra.algHom = IsScalarTower.toAlgHom

theorem Algebra.algHom_apply (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (y : S) : (IsScalarTower.toAlgHom R S A) y = (algebraMap S A) y

Alias of IsScalarTower.toAlgHom_apply.

@[simp]

theorem AlgHomClass.toRingHom_ofClass {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 : Type u_4} [FunLike F A B] [AlgHomClass F R A B] (f : F) : ↑↑f = ↑f

@[deprecated AlgHomClass.toRingHom_ofClass (since := "2026-09-08")]

theorem AlgHomClass.toRingHom_toAlgHom {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 : Type u_4} [FunLike F A B] [AlgHomClass F R A B] (f : F) : ↑↑f = ↑f

Alias of AlgHomClass.toRingHom_ofClass.

def RingHom.toNatAlgHom {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S

Reinterpret a RingHom as an ℕ-algebra homomorphism.

Equations

• f.toNatAlgHom = { toFun := ⇑f, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem RingHom.toNatAlgHom_coe {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) : ⇑f.toNatAlgHom = ⇑f

theorem RingHom.toNatAlgHom_apply {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] (f : R →+* S) (x : R) : f.toNatAlgHom x = f x

def RingHom.equivNatAlgHom (R : Type u_1) (S : Type u_2) [Semiring R] [Semiring S] : (R →+* S) ≃ (R →ₐ[ℕ] S)

Ring homomorphisms are the same as ℕ-algebra homomorphisms.

Equations

• RingHom.equivNatAlgHom R S = { toFun := RingHom.toNatAlgHom, invFun := AlgHom.toRingHom, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem RingHom.equivNatAlgHom_symm_apply (R : Type u_1) (S : Type u_2) [Semiring R] [Semiring S] (self : R →ₐ[ℕ] S) : (equivNatAlgHom R S).symm self = self.toRingHom

@[simp]

theorem RingHom.equivNatAlgHom_apply (R : Type u_1) (S : Type u_2) [Semiring R] [Semiring S] (f : R →+* S) : (equivNatAlgHom R S) f = f.toNatAlgHom

def RingHom.toIntAlgHom {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) : R →ₐ[ℤ] S

Reinterpret a RingHom as a ℤ-algebra homomorphism.

Equations

• f.toIntAlgHom = { toRingHom := f, commutes' := ⋯ }

@[simp]

theorem RingHom.toIntAlgHom_coe {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) : ⇑f.toIntAlgHom = ⇑f

theorem RingHom.toIntAlgHom_apply {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (x : R) : f.toIntAlgHom x = f x

theorem RingHom.toIntAlgHom_injective {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] : Function.Injective toIntAlgHom

def RingHom.equivIntAlgHom (R : Type u_1) (S : Type u_2) [Ring R] [Ring S] : (R →+* S) ≃ (R →ₐ[ℤ] S)

Ring homomorphisms are the same as ℤ-algebra homomorphisms.

Equations

• RingHom.equivIntAlgHom R S = { toFun := RingHom.toIntAlgHom, invFun := AlgHom.toRingHom, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem RingHom.equivIntAlgHom_symm_apply (R : Type u_1) (S : Type u_2) [Ring R] [Ring S] (self : R →ₐ[ℤ] S) : (equivIntAlgHom R S).symm self = self.toRingHom

@[simp]

theorem RingHom.equivIntAlgHom_apply (R : Type u_1) (S : Type u_2) [Ring R] [Ring S] (f : R →+* S) : (equivIntAlgHom R S) f = f.toIntAlgHom

def Algebra.ofId (R : Type u) (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : R →ₐ[R] A

AlgebraMap as an AlgHom.

Equations

• Algebra.ofId R A = { toRingHom := algebraMap R A, commutes' := ⋯ }

@[simp]

theorem Algebra.ofId_self {R : Type u} [CommSemiring R] : ofId R R = AlgHom.id R R

@[simp]

theorem Algebra.toRingHom_ofId {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : ↑(ofId R A) = algebraMap R A

@[simp]

theorem Algebra.ofId_apply {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (r : R) : (ofId R A) r = (algebraMap R A) r

instance Algebra.subsingleton_id {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : Subsingleton (R →ₐ[R] A)

This is a special case of a more general instance that we define in a later file.

theorem Algebra.ext_id {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (f g : R →ₐ[R] A) : f = g

This ext lemma closes trivial subgoals created when chaining heterobasic ext lemmas.

theorem Algebra.ext_id_iff {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] {f g : R →ₐ[R] A} : f = g ↔ True

@[simp]

theorem Algebra.comp_ofId {R : Type u} (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (φ : A →ₐ[R] B) : φ.comp (ofId R A) = ofId R B

@[instance_reducible]

instance Algebra.instMulDistribMulActionAlgHomUnits {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] : MulDistribMulAction (A →ₐ[R] A) Aˣ

Equations

• Algebra.instMulDistribMulActionAlgHomUnits A = { smul := fun (f : A →ₐ[R] A) => ⇑(Units.map ↑f), mul_smul := ⋯, one_smul := ⋯, smul_one := ⋯, smul_mul := ⋯ }

@[simp]

theorem Algebra.smul_units_def {R : Type u} (A : Type v) [CommSemiring R] [Semiring A] [Algebra R A] (f : A →ₐ[R] A) (x : Aˣ) : f • x = (Units.map ↑f) x

theorem Algebra.algebraMapSubmonoid_map_eq {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (M : Submonoid R) {B : Type w} [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : Submonoid.map f (algebraMapSubmonoid A M) = algebraMapSubmonoid B M

theorem Algebra.algebraMapSubmonoid_le_comap {R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [Algebra R A] (M : Submonoid R) {B : Type w} [Semiring B] [Algebra R B] (f : A →ₐ[R] B) : algebraMapSubmonoid A M ≤ Submonoid.comap f.toRingHom (algebraMapSubmonoid B M)

def MulSemiringAction.toAlgHom {M : Type u_1} (R : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] (m : M) : A →ₐ[R] A

Each element of the monoid defines an algebra homomorphism.

This is a stronger version of MulSemiringAction.toRingHom and DistribSMul.toLinearMap.

Equations

• MulSemiringAction.toAlgHom R A m = { toFun := fun (a : A) => m • a, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem MulSemiringAction.toAlgHom_apply {M : Type u_1} (R : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] (m : M) (a : A) : (toAlgHom R A m) a = m • a

theorem MulSemiringAction.toAlgHom_injective {M : Type u_1} (R : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] [MulSemiringAction M A] [SMulCommClass M R A] [FaithfulSMul M A] : Function.Injective (toAlgHom R A)

@[instance_reducible]

instance uniqueOfRight {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommSemiring R] [Semiring S] [Semiring T] [Algebra R S] [Algebra R T] [Subsingleton T] : Unique (S →ₐ[R] T)

Equations

• uniqueOfRight = { default := AlgHom.ofLinearMap default ⋯ ⋯, uniq := ⋯ }

@[simp]

theorem AlgHom.default_apply {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommSemiring R] [Semiring S] [Semiring T] [Algebra R S] [Algebra R T] [Subsingleton T] (x : S) : default x = 0