Multiplication on the left/right as additive automorphisms

In this file we define AddAut.mulLeft and AddAut.mulRight.

See also AddMonoidHom.mulLeft, AddMonoidHom.mulRight, AddMonoid.End.mulLeft, and AddMonoid.End.mulRight for multiplication by R as an endomorphism instead of multiplication by Rˣ as an automorphism.

def AddAut.mulLeft {R : Type u_1} [Semiring R] : Rˣ →* AddAut R

Left multiplication by a unit of a semiring as an additive automorphism.

Equations

• AddAut.mulLeft = DistribMulAction.toAddAut Rˣ R

@[simp]

theorem AddAut.mulLeft_apply_symm_apply {R : Type u_1} [Semiring R] (x : Rˣ) (a✝ : R) : (AddEquiv.symm (mulLeft x)) a✝ = x⁻¹ • a✝

@[simp]

theorem AddAut.mulLeft_apply_apply {R : Type u_1} [Semiring R] (x : Rˣ) (a✝ : R) : (mulLeft x) a✝ = x • a✝

def AddAut.mulRight {R : Type u_1} [Semiring R] (u : Rˣ) : AddAut R

Right multiplication by a unit of a semiring as an additive automorphism.

Equations

• AddAut.mulRight u = (DistribMulAction.toAddAut Rᵐᵒᵖˣ R) (Units.opEquiv.symm (MulOpposite.op u))

@[simp]

theorem AddAut.mulRight_apply {R : Type u_1} [Semiring R] (u : Rˣ) (x : R) : (mulRight u) x = x * ↑u

@[simp]

theorem AddAut.mulRight_symm_apply {R : Type u_1} [Semiring R] (u : Rˣ) (x : R) : (AddEquiv.symm (mulRight u)) x = x * ↑u⁻¹