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Mathlib.Algebra.Ring.AddAut

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 as an automorphism.

@[simp]
theorem AddAut.mulLeft_apply_symm_apply {R : Type u_1} [Semiring R] (x : Rˣ) :
∀ (a : R), (AddEquiv.symm (AddAut.mulLeft x)) a = x⁻¹ a
@[simp]
theorem AddAut.mulLeft_apply_apply {R : Type u_1} [Semiring R] (x : Rˣ) :
∀ (a : R), (AddAut.mulLeft x) a = x a
def AddAut.mulLeft {R : Type u_1} [Semiring R] :

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

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

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

    Equations
    Instances For
      @[simp]
      theorem AddAut.mulRight_apply {R : Type u_1} [Semiring R] (u : Rˣ) (x : R) :
      (AddAut.mulRight u) x = x * u
      @[simp]
      theorem AddAut.mulRight_symm_apply {R : Type u_1} [Semiring R] (u : Rˣ) (x : R) :