mathlib documentation

algebra.group_ring_action

Group action on rings #

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

Note that algebra does not satisfy the axioms of mul_semiring_action.

Implementation notes #

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

Tags #

group action, invariant subring

source
@[class]
structure mul_semiring_action (M : Type u) (R : Type v) [monoid M] [semiring R] :
Type (max u v)

Typeclass for multiplicative actions by monoids on semirings.

This combines distrib_mul_action with mul_distrib_mul_action.

Instances
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@[protected, instance]
def mul_semiring_action.to_mul_distrib_mul_action (M : Type u) [monoid M] (R : Type v) [semiring R] [h : mul_semiring_action M R] :
mul_distrib_mul_action M R
Equations
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@[simp]
theorem mul_semiring_action.to_ring_hom_apply (M : Type u) [monoid M] (R : Type v) [semiring R] [mul_semiring_action M R] (x : M) (ᾰ : R) :
⇑(mul_semiring_action.to_ring_hom M R x) ᾰ = (mul_distrib_mul_action.to_monoid_hom R x).to_fun ᾰ
source
def mul_semiring_action.to_ring_hom (M : Type u) [monoid M] (R : Type v) [semiring R] [mul_semiring_action M R] (x : M) :
R →+* R

Each element of the monoid defines a semiring homomorphism.

Equations
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theorem to_ring_hom_injective (M : Type u) [monoid M] (R : Type v) [semiring R] [mul_semiring_action M R] [has_faithful_scalar M R] :
function.injective (mul_semiring_action.to_ring_hom M R)
source
@[simp]
theorem mul_semiring_action.to_ring_equiv_apply (G : Type u) [group G] (R : Type v) [semiring R] [mul_semiring_action G R] (x : G) (ᾰ : R) :
⇑(mul_semiring_action.to_ring_equiv G R x) ᾰ = (distrib_mul_action.to_add_equiv R x).to_fun ᾰ
source
def mul_semiring_action.to_ring_equiv (G : Type u) [group G] (R : Type v) [semiring R] [mul_semiring_action G R] (x : G) :
R ≃+* R

Each element of the group defines a semiring isomorphism.

Equations
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@[simp]
theorem mul_semiring_action.to_ring_equiv_symm_apply (G : Type u) [group G] (R : Type v) [semiring R] [mul_semiring_action G R] (x : G) (ᾰ : R) :
⇑((mul_semiring_action.to_ring_equiv G R x).symm) ᾰ = (distrib_mul_action.to_add_equiv R x).inv_fun ᾰ
source
@[protected, instance]
def submonoid.mul_semiring_action {M : Type u} [monoid M] {R : Type v} [semiring R] [mul_semiring_action M R] (H : submonoid M) :
mul_semiring_action ↥H R

A stronger version of submonoid.distrib_mul_action.

Equations
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@[protected, instance]
def subgroup.mul_semiring_action {G : Type u} [group G] {R : Type v} [semiring R] [mul_semiring_action G R] (H : subgroup G) :
mul_semiring_action ↥H R

A stronger version of subgroup.distrib_mul_action.

Equations
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@[protected, instance]
def subsemiring.mul_semiring_action {R : Type v} [semiring R] {R' : Type u_1} [semiring R'] [mul_semiring_action R' R] (H : subsemiring R') :
mul_semiring_action ↥H R

A stronger version of subsemiring.distrib_mul_action.

Equations
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@[protected, instance]
def subring.mul_semiring_action {R : Type v} [semiring R] {R' : Type u_1} [ring R'] [mul_semiring_action R' R] (H : subring R') :
mul_semiring_action ↥H R

A stronger version of subring.distrib_mul_action.

Equations
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@[simp]
theorem smul_inv'' {M : Type u} [monoid M] {F : Type v} [division_ring F] [mul_semiring_action M F] (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.

source
@[class]
structure is_invariant_subring (M : Type u) [monoid M] {R : Type v} [ring R] [mul_semiring_action M R] (S : subring R) :
Prop
  • smul_mem : ∀ (m : M) {x : R}, x ∈ S → m • x ∈ S

A typeclass for subrings invariant under a mul_semiring_action.

source
@[protected, instance]
def is_invariant_subring.to_mul_semiring_action (M : Type u) [monoid M] {R : Type v} [ring R] [mul_semiring_action M R] (S : subring R) [is_invariant_subring M S] :
mul_semiring_action M ↥S
Equations