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) [Monoid M] [Semiring R] extends DistribMulAction :
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
    Equations
    @[simp]
    theorem MulSemiringAction.toRingHom_apply (M : Type u_1) [Monoid M] (R : Type v) [Semiring R] [MulSemiringAction M R] (x : M) :
    ∀ (x : R), ↑(MulSemiringAction.toRingHom M R x) x = x x
    def MulSemiringAction.toRingHom (M : Type u_1) [Monoid M] (R : Type v) [Semiring R] [MulSemiringAction M R] (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.
    Instances For
      @[simp]
      theorem MulSemiringAction.toRingEquiv_apply (G : Type u_3) [Group G] (R : Type v) [Semiring R] [MulSemiringAction G R] (x : G) :
      ∀ (a : R), ↑(MulSemiringAction.toRingEquiv G R x) a = x a
      @[simp]
      theorem MulSemiringAction.toRingEquiv_symm_apply (G : Type u_3) [Group G] (R : Type v) [Semiring R] [MulSemiringAction G R] (x : G) :
      ∀ (a : R), ↑(RingEquiv.symm (MulSemiringAction.toRingEquiv G R x)) a = x⁻¹ a
      def MulSemiringAction.toRingEquiv (G : Type u_3) [Group G] (R : Type v) [Semiring R] [MulSemiringAction G R] (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.
      Instances For
        @[reducible]
        def MulSemiringAction.compHom {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] (R : Type v) [Semiring R] (f : N →* M) [MulSemiringAction M R] :

        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.
        Instances For
          @[simp]
          theorem smul_inv'' {M : Type u_1} [Monoid M] {F : Type v} [DivisionRing F] [MulSemiringAction 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.