# 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) [inst : ] [inst : ] extends :
Type (maxuv)
• Multipliying 1 by a scalar gives 1

smul_one : ∀ (g : M), g 1 = 1
• Scalara multiplication distributes across multiplication

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

Typeclass for multiplicative actions by monoids on semirings.

This combines DistribMulAction with MulDistribMulAction.

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

Each element of the monoid defines a semiring homomorphism.

Equations
• One or more equations did not get rendered due to their size.
theorem toRingHom_injective (M : Type u_1) [inst : ] (R : Type v) [inst : ] [inst : ] [inst : ] :
@[simp]
theorem MulSemiringAction.toRingEquiv_apply (G : Type u_1) [inst : ] (R : Type v) [inst : ] [inst : ] (x : G) :
∀ (a : R), ↑() a = x a
@[simp]
theorem MulSemiringAction.toRingEquiv_symm_apply (G : Type u_1) [inst : ] (R : Type v) [inst : ] [inst : ] (x : G) :
∀ (a : R), ↑() a = x⁻¹ a
def MulSemiringAction.toRingEquiv (G : Type u_1) [inst : ] (R : Type v) [inst : ] [inst : ] (x : G) :
R ≃+* R

Each element of the group defines a semiring isomorphism.

Equations
• One or more equations did not get rendered due to their size.
def MulSemiringAction.compHom {M : Type u_1} {N : Type u_2} [inst : ] [inst : ] (R : Type v) [inst : ] (f : N →* M) [inst : ] :

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem smul_inv'' {M : Type u_1} [inst : ] {F : Type v} [inst : ] [inst : ] (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.