# Documentation

Mathlib.Algebra.GroupRingAction.Basic

# Group action on rings #

This file defines the typeclass of monoid acting on semirings MulSemiringAction M R, and the corresponding typeclass of invariant subrings.

Note that Algebra does not satisfy the axioms of MulSemiringAction.

## Implementation notes #

There is no separate typeclass for group acting on rings, group acting on fields, etc. They are all grouped under MulSemiringAction.

## Tags #

group action, invariant subring

class MulSemiringAction (M : Type u) (R : Type v) [] [] extends :
Type (max u v)
• smul : MRR
• one_smul : ∀ (b : R), 1 b = b
• mul_smul : ∀ (x y : M) (b : R), (x * y) b = x y b
• smul_zero : ∀ (a : M), a 0 = 0
• smul_add : ∀ (a : M) (x y : R), a (x + y) = a x + a y
• smul_one : ∀ (g : M), g 1 = 1

Multipliying 1 by a scalar gives 1

• smul_mul : ∀ (g : M) (x y : R), g (x * y) = g x * g y

Scalar multiplication distributes across multiplication

Typeclass for multiplicative actions by monoids on semirings.

This combines DistribMulAction with MulDistribMulAction.

Instances
instance MulSemiringAction.toMulDistribMulAction (M : Type u_1) [] (R : Type v) [] [h : ] :
@[simp]
theorem MulSemiringAction.toRingHom_apply (M : Type u_1) [] (R : Type v) [] [] (x : M) :
∀ (x : R), ↑() x = x x
def MulSemiringAction.toRingHom (M : Type u_1) [] (R : Type v) [] [] (x : M) :
R →+* R

Each element of the monoid defines a semiring homomorphism.

Instances For
theorem toRingHom_injective (M : Type u_1) [] (R : Type v) [] [] [] :
@[simp]
theorem MulSemiringAction.toRingEquiv_apply (G : Type u_3) [] (R : Type v) [] [] (x : G) :
∀ (a : R), ↑() a = x a
@[simp]
theorem MulSemiringAction.toRingEquiv_symm_apply (G : Type u_3) [] (R : Type v) [] [] (x : G) :
∀ (a : R), ↑() a = x⁻¹ a
def MulSemiringAction.toRingEquiv (G : Type u_3) [] (R : Type v) [] [] (x : G) :
R ≃+* R

Each element of the group defines a semiring isomorphism.

Instances For
@[reducible]
def MulSemiringAction.compHom {M : Type u_1} {N : Type u_2} [] [] (R : Type v) [] (f : N →* M) [] :

Compose a MulSemiringAction with a MonoidHom, with action f r' • m. See note [reducible non-instances].

Instances For
@[simp]
theorem smul_inv'' {M : Type u_1} [] {F : Type v} [] [] (x : M) (m : F) :
x m⁻¹ = (x m)⁻¹

Note that smul_inv' refers to the group case, and smul_inv has an additional inverse on x.