Centroid homomorphisms

Let A be a (nonunital, non-associative) algebra. The centroid of A is the set of linear maps T on A such that T commutes with left and right multiplication, that is to say, for all a and b in A, $$ T(ab) = (Ta)b, T(ab) = a(Tb). $$ In mathlib we call elements of the centroid "centroid homomorphisms" (CentroidHom) in keeping with AddMonoidHom etc.

We use the DFunLike design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.

Types of morphisms

• CentroidHom: Maps which preserve left and right multiplication.

Typeclasses

• CentroidHomClass

References

• [Jacobson, Structure of Rings][Jacobson1956]
• [McCrimmon, A taste of Jordan algebras][mccrimmon2004]

Tags

centroid

structure CentroidHom (α : Type u_6) [NonUnitalNonAssocSemiring α] extends α →+ α : Type u_6

The type of centroid homomorphisms from α to α.

• toFun : α → α
• map_zero' : (↑self.toAddMonoidHom).toFun 0 = 0
• map_add' (x y : α) : (↑self.toAddMonoidHom).toFun (x + y) = (↑self.toAddMonoidHom).toFun x + (↑self.toAddMonoidHom).toFun y
• map_mul_left' (a b : α) : (↑self.toAddMonoidHom).toFun (a * b) = a * (↑self.toAddMonoidHom).toFun b

  Commutativity of centroid homomorphisms with left multiplication.

• map_mul_right' (a b : α) : (↑self.toAddMonoidHom).toFun (a * b) = (↑self.toAddMonoidHom).toFun a * b

  Commutativity of centroid homomorphisms with right multiplication.

class CentroidHomClass (F : Type u_6) (α : outParam (Type u_7)) [NonUnitalNonAssocSemiring α] [FunLike F α α] extends AddMonoidHomClass F α α : Prop

CentroidHomClass F α states that F is a type of centroid homomorphisms.

You should extend this class when you extend CentroidHom.

• map_add (f : F) (x y : α) : f (x + y) = f x + f y
• map_zero (f : F) : f 0 = 0
• map_mul_left (f : F) (a b : α) : f (a * b) = a * f b

  Commutativity of centroid homomorphisms with left multiplication.

• map_mul_right (f : F) (a b : α) : f (a * b) = f a * b

  Commutativity of centroid homomorphisms with right multiplication.

@[implicit_reducible]

instance instCoeTCCentroidHomOfCentroidHomClass {F : Type u_1} {α : Type u_5} [NonUnitalNonAssocSemiring α] [FunLike F α α] [CentroidHomClass F α] : CoeTC F (CentroidHom α)

Centroid homomorphisms

@[implicit_reducible]

instance CentroidHom.instFunLike {α : Type u_5} [NonUnitalNonAssocSemiring α] : FunLike (CentroidHom α) α α

instance CentroidHom.instCentroidHomClass {α : Type u_5} [NonUnitalNonAssocSemiring α] : CentroidHomClass (CentroidHom α) α

theorem CentroidHom.toFun_eq_coe {α : Type u_5} [NonUnitalNonAssocSemiring α] {f : CentroidHom α} : (↑f.toAddMonoidHom).toFun = ⇑f

theorem CentroidHom.ext {α : Type u_5} [NonUnitalNonAssocSemiring α] {f g : CentroidHom α} (h : ∀ (a : α), f a = g a) : f = g

theorem CentroidHom.ext_iff {α : Type u_5} [NonUnitalNonAssocSemiring α] {f g : CentroidHom α} : f = g ↔ ∀ (a : α), f a = g a

@[simp]

theorem CentroidHom.coe_toAddMonoidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : ⇑↑f = ⇑f

@[simp]

theorem CentroidHom.toAddMonoidHom_eq_coe {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : f.toAddMonoidHom = ↑f

theorem CentroidHom.coe_toAddMonoidHom_injective {α : Type u_5} [NonUnitalNonAssocSemiring α] : Function.Injective AddMonoidHomClass.toAddMonoidHom

def CentroidHom.toEnd {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : AddMonoid.End α

Turn a centroid homomorphism into an additive monoid endomorphism.

theorem CentroidHom.toEnd_injective {α : Type u_5} [NonUnitalNonAssocSemiring α] : Function.Injective toEnd

def CentroidHom.copy {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (f' : α → α) (h : f' = ⇑f) : CentroidHom α

Copy of a CentroidHom with a new toFun equal to the old one. Useful to fix definitional equalities.

@[simp]

theorem CentroidHom.coe_copy {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (f' : α → α) (h : f' = ⇑f) : ⇑(f.copy f' h) = f'

theorem CentroidHom.copy_eq {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (f' : α → α) (h : f' = ⇑f) : f.copy f' h = f

def CentroidHom.id (α : Type u_5) [NonUnitalNonAssocSemiring α] : CentroidHom α

id as a CentroidHom.

@[implicit_reducible]

instance CentroidHom.instInhabited (α : Type u_5) [NonUnitalNonAssocSemiring α] : Inhabited (CentroidHom α)

@[simp]

theorem CentroidHom.coe_id (α : Type u_5) [NonUnitalNonAssocSemiring α] : ⇑(CentroidHom.id α) = id

@[simp]

theorem CentroidHom.toAddMonoidHom_id (α : Type u_5) [NonUnitalNonAssocSemiring α] : ↑(CentroidHom.id α) = AddMonoidHom.id α

@[simp]

theorem CentroidHom.id_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : (CentroidHom.id α) a = a

def CentroidHom.comp {α : Type u_5} [NonUnitalNonAssocSemiring α] (g f : CentroidHom α) : CentroidHom α

Composition of CentroidHoms as a CentroidHom.

@[simp]

theorem CentroidHom.coe_comp {α : Type u_5} [NonUnitalNonAssocSemiring α] (g f : CentroidHom α) : ⇑(g.comp f) = ⇑g ∘ ⇑f

@[simp]

theorem CentroidHom.comp_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (g f : CentroidHom α) (a : α) : (g.comp f) a = g (f a)

@[simp]

theorem CentroidHom.coe_comp_addMonoidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] (g f : CentroidHom α) : ↑(g.comp f) = (↑g).comp ↑f

@[simp]

theorem CentroidHom.comp_assoc {α : Type u_5} [NonUnitalNonAssocSemiring α] (h g f : CentroidHom α) : (h.comp g).comp f = h.comp (g.comp f)

@[simp]

theorem CentroidHom.comp_id {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : f.comp (CentroidHom.id α) = f

@[simp]

theorem CentroidHom.id_comp {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : (CentroidHom.id α).comp f = f

@[simp]

theorem CentroidHom.cancel_right {α : Type u_5} [NonUnitalNonAssocSemiring α] {g₁ g₂ f : CentroidHom α} (hf : Function.Surjective ⇑f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂

@[simp]

theorem CentroidHom.cancel_left {α : Type u_5} [NonUnitalNonAssocSemiring α] {g f₁ f₂ : CentroidHom α} (hg : Function.Injective ⇑g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

@[implicit_reducible]

instance CentroidHom.instZero {α : Type u_5} [NonUnitalNonAssocSemiring α] : Zero (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instOne {α : Type u_5} [NonUnitalNonAssocSemiring α] : One (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instAdd {α : Type u_5} [NonUnitalNonAssocSemiring α] : Add (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instMul {α : Type u_5} [NonUnitalNonAssocSemiring α] : Mul (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instSMul {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] : SMul M (CentroidHom α)

instance CentroidHom.instIsScalarTower {M : Type u_2} {N : Type u_3} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [Monoid N] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α] [SMul M N] [IsScalarTower M N α] : IsScalarTower M N (CentroidHom α)

instance CentroidHom.instSMulCommClass {M : Type u_2} {N : Type u_3} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [Monoid N] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α] [SMulCommClass M N α] : SMulCommClass M N (CentroidHom α)

instance CentroidHom.instIsCentralScalar {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] [DistribMulAction Mᵐᵒᵖ α] [IsCentralScalar M α] : IsCentralScalar M (CentroidHom α)

instance CentroidHom.isScalarTowerRight {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] : IsScalarTower M (CentroidHom α) (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.hasNPowNat {α : Type u_5} [NonUnitalNonAssocSemiring α] : Pow (CentroidHom α) ℕ

@[simp]

theorem CentroidHom.coe_zero {α : Type u_5} [NonUnitalNonAssocSemiring α] : ⇑0 = 0

@[simp]

theorem CentroidHom.coe_one {α : Type u_5} [NonUnitalNonAssocSemiring α] : ⇑1 = id

@[simp]

theorem CentroidHom.coe_add {α : Type u_5} [NonUnitalNonAssocSemiring α] (f g : CentroidHom α) : ⇑(f + g) = ⇑f + ⇑g

@[simp]

theorem CentroidHom.coe_mul {α : Type u_5} [NonUnitalNonAssocSemiring α] (f g : CentroidHom α) : ⇑(f * g) = ⇑f ∘ ⇑g

@[simp]

theorem CentroidHom.coe_smul {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] (n : M) (f : CentroidHom α) : ⇑(n • f) = n • ⇑f

@[simp]

theorem CentroidHom.zero_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : 0 a = 0

@[simp]

theorem CentroidHom.one_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : 1 a = a

@[simp]

theorem CentroidHom.add_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (f g : CentroidHom α) (a : α) : (f + g) a = f a + g a

@[simp]

theorem CentroidHom.mul_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (f g : CentroidHom α) (a : α) : (f * g) a = f (g a)

@[simp]

theorem CentroidHom.smul_apply {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] (n : M) (f : CentroidHom α) (a : α) : (n • f) a = n • f a

@[simp]

theorem CentroidHom.toEnd_zero {α : Type u_5} [NonUnitalNonAssocSemiring α] : toEnd 0 = 0

@[simp]

theorem CentroidHom.toEnd_add {α : Type u_5} [NonUnitalNonAssocSemiring α] (x y : CentroidHom α) : (x + y).toEnd = x.toEnd + y.toEnd

theorem CentroidHom.toEnd_smul {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] (m : M) (x : CentroidHom α) : (m • x).toEnd = m • x.toEnd

@[implicit_reducible]

instance CentroidHom.instAddCommMonoid {α : Type u_5} [NonUnitalNonAssocSemiring α] : AddCommMonoid (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instNatCast {α : Type u_5} [NonUnitalNonAssocSemiring α] : NatCast (CentroidHom α)

@[simp]

theorem CentroidHom.coe_natCast {α : Type u_5} [NonUnitalNonAssocSemiring α] (n : ℕ) : ⇑↑n = n • ⇑(CentroidHom.id α)

theorem CentroidHom.natCast_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (n : ℕ) (m : α) : ↑n m = n • m

@[simp]

theorem CentroidHom.toEnd_one {α : Type u_5} [NonUnitalNonAssocSemiring α] : toEnd 1 = 1

@[simp]

theorem CentroidHom.toEnd_mul {α : Type u_5} [NonUnitalNonAssocSemiring α] (x y : CentroidHom α) : (x * y).toEnd = x.toEnd * y.toEnd

@[simp]

theorem CentroidHom.toEnd_pow {α : Type u_5} [NonUnitalNonAssocSemiring α] (x : CentroidHom α) (n : ℕ) : (x ^ n).toEnd = x.toEnd ^ n

@[simp]

theorem CentroidHom.toEnd_natCast {α : Type u_5} [NonUnitalNonAssocSemiring α] (n : ℕ) : (↑n).toEnd = ↑n

@[implicit_reducible]

instance CentroidHom.instSemiring {α : Type u_5} [NonUnitalNonAssocSemiring α] : Semiring (CentroidHom α)

def CentroidHom.toEndRingHom (α : Type u_5) [NonUnitalNonAssocSemiring α] : CentroidHom α →+* AddMonoid.End α

CentroidHom.toEnd as a RingHom.

@[simp]

theorem CentroidHom.toEndRingHom_apply (α : Type u_5) [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : (toEndRingHom α) f = f.toEnd

theorem CentroidHom.comp_mul_comm {α : Type u_5} [NonUnitalNonAssocSemiring α] (T S : CentroidHom α) (a b : α) : (⇑T ∘ ⇑S) (a * b) = (⇑S ∘ ⇑T) (a * b)

@[implicit_reducible]

instance CentroidHom.instDistribMulAction {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] : DistribMulAction M (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instModule {R : Type u_4} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Semiring R] [Module R α] [SMulCommClass R α α] [IsScalarTower R α α] : Module R (CentroidHom α)

The following instances show that α is a non-unital and non-associative algebra over CentroidHom α.

@[implicit_reducible]

instance CentroidHom.applyModule {α : Type u_5} [NonUnitalNonAssocSemiring α] : Module (CentroidHom α) α

The tautological action by CentroidHom α on α.

This generalizes Function.End.applyMulAction.

@[simp]

theorem CentroidHom.smul_def {α : Type u_5} [NonUnitalNonAssocSemiring α] (T : CentroidHom α) (a : α) : T • a = T a

instance CentroidHom.instSMulCommClass_1 {α : Type u_5} [NonUnitalNonAssocSemiring α] : SMulCommClass (CentroidHom α) α α

instance CentroidHom.instSMulCommClass_2 {α : Type u_5} [NonUnitalNonAssocSemiring α] : SMulCommClass α (CentroidHom α) α

instance CentroidHom.instIsScalarTower_1 {α : Type u_5} [NonUnitalNonAssocSemiring α] : IsScalarTower (CentroidHom α) α α

Let α be an algebra over R, such that the canonical ring homomorphism of R into CentroidHom α lies in the center of CentroidHom α. Then CentroidHom α is an algebra over R

def Module.toCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] {R : Type u_6} [CommSemiring R] [Module R α] [SMulCommClass R α α] [IsScalarTower R α α] : R →+* CentroidHom α

The natural ring homomorphism from R into CentroidHom α.

This is a stronger version of Module.toAddMonoidEnd.

@[simp]

theorem Module.toCentroidHom_apply_toFun {α : Type u_5} [NonUnitalNonAssocSemiring α] {R : Type u_6} [CommSemiring R] [Module R α] [SMulCommClass R α α] [IsScalarTower R α α] (x : R) (a : α) : (toCentroidHom x) a = x • a

theorem CentroidHom.centroid_eq_centralizer_mulLeftRight {α : Type u_5} [NonUnitalNonAssocSemiring α] : (toEndRingHom α).rangeS = Subsemiring.centralizer (Set.range ⇑AddMonoid.End.mulLeft ∪ Set.range ⇑AddMonoid.End.mulRight)

def CentroidHom.centerToCentroidCenter {α : Type u_5} [NonUnitalNonAssocSemiring α] : ↥(NonUnitalSubsemiring.center α) →ₙ+* ↥(Subsemiring.center (CentroidHom α))

The canonical homomorphism from the center into the center of the centroid

@[implicit_reducible]

instance CentroidHom.instFunLikeSubtypeMemSubsemiringCenter {α : Type u_5} [NonUnitalNonAssocSemiring α] : FunLike (↥(Subsemiring.center (CentroidHom α))) α α

theorem CentroidHom.centerToCentroidCenter_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (z : ↥(NonUnitalSubsemiring.center α)) (a : α) : (centerToCentroidCenter z) a = ↑z * a

def CentroidHom.centerToCentroid {α : Type u_5} [NonUnitalNonAssocSemiring α] : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α

The canonical homomorphism from the center into the centroid

theorem CentroidHom.centerToCentroid_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (z : ↥(NonUnitalSubsemiring.center α)) (a : α) : (centerToCentroid z) a = ↑z * a

theorem NonUnitalNonAssocSemiring.mem_center_iff {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : a ∈ NonUnitalSubsemiring.center α ↔ AddMonoid.End.mulRight a = AddMonoid.End.mulLeft a ∧ AddMonoid.End.mulLeft a ∈ (CentroidHom.toEndRingHom α).rangeS

theorem NonUnitalNonAssocCommSemiring.mem_center_iff {α : Type u_5} [NonUnitalNonAssocCommSemiring α] (a : α) : a ∈ NonUnitalSubsemiring.center α ↔ ∀ (b : α), Commute (AddMonoid.End.mulLeft b) (AddMonoid.End.mulLeft a)

def CentroidHom.centerIsoCentroid {α : Type u_5} [NonAssocSemiring α] : ↥(Subsemiring.center α) ≃+* CentroidHom α

The canonical isomorphism from the center of a (non-associative) semiring onto its centroid.

@[implicit_reducible]

instance CentroidHom.instNeg {α : Type u_5} [NonUnitalNonAssocRing α] : Neg (CentroidHom α)

Negation of CentroidHoms as a CentroidHom.

@[implicit_reducible]

instance CentroidHom.instSub {α : Type u_5} [NonUnitalNonAssocRing α] : Sub (CentroidHom α)

@[implicit_reducible]

instance CentroidHom.instIntCast {α : Type u_5} [NonUnitalNonAssocRing α] : IntCast (CentroidHom α)

@[simp]

theorem CentroidHom.coe_intCast {α : Type u_5} [NonUnitalNonAssocRing α] (z : ℤ) : ⇑↑z = z • ⇑(CentroidHom.id α)

theorem CentroidHom.intCast_apply {α : Type u_5} [NonUnitalNonAssocRing α] (z : ℤ) (m : α) : ↑z m = z • m

@[simp]

theorem CentroidHom.toEnd_neg {α : Type u_5} [NonUnitalNonAssocRing α] (x : CentroidHom α) : (-x).toEnd = -x.toEnd

@[simp]

theorem CentroidHom.toEnd_sub {α : Type u_5} [NonUnitalNonAssocRing α] (x y : CentroidHom α) : (x - y).toEnd = x.toEnd - y.toEnd

@[implicit_reducible]

instance CentroidHom.instAddCommGroup {α : Type u_5} [NonUnitalNonAssocRing α] : AddCommGroup (CentroidHom α)

@[simp]

theorem CentroidHom.coe_neg {α : Type u_5} [NonUnitalNonAssocRing α] (f : CentroidHom α) : ⇑(-f) = -⇑f

@[simp]

theorem CentroidHom.coe_sub {α : Type u_5} [NonUnitalNonAssocRing α] (f g : CentroidHom α) : ⇑(f - g) = ⇑f - ⇑g

@[simp]

theorem CentroidHom.neg_apply {α : Type u_5} [NonUnitalNonAssocRing α] (f : CentroidHom α) (a : α) : (-f) a = -f a

@[simp]

theorem CentroidHom.sub_apply {α : Type u_5} [NonUnitalNonAssocRing α] (f g : CentroidHom α) (a : α) : (f - g) a = f a - g a

@[simp]

theorem CentroidHom.toEnd_intCast {α : Type u_5} [NonUnitalNonAssocRing α] (z : ℤ) : (↑z).toEnd = ↑z

@[implicit_reducible]

instance CentroidHom.instRing {α : Type u_5} [NonUnitalNonAssocRing α] : Ring (CentroidHom α)

@[reducible, inline]

abbrev CentroidHom.commRing {α : Type u_5} [NonUnitalRing α] (h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α)

A prime associative ring has commutative centroid.