Subrings invariant under an action

class IsInvariantSubring (M : Type u_1) {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (S : Subring R) : Prop

A typeclass for subrings invariant under a MulSemiringAction.

• smul_mem : ∀ (m : M) {x : R}, x ∈ S → m • x ∈ S

Instances

• instIsInvariantSubringOfIsInvariantSubfield

theorem IsInvariantSubring.smul_mem {M : Type u_1} {R : Type u_2} : ∀ {inst : Monoid M} {inst_1 : Ring R} {inst_2 : MulSemiringAction M R} {S : Subring R} [self : IsInvariantSubring M S] (m : M) {x : R}, x ∈ S → m • x ∈ S

instance IsInvariantSubring.toMulSemiringAction (M : Type u_1) {R : Type u_2} [Monoid M] [Ring R] [MulSemiringAction M R] (S : Subring R) [IsInvariantSubring M S] : MulSemiringAction M ↥S

Equations

• IsInvariantSubring.toMulSemiringAction M S = MulSemiringAction.mk ⋯ ⋯

def IsInvariantSubring.subtypeHom (M : Type u_1) [Monoid M] {R' : Type u_2} [Ring R'] [MulSemiringAction M R'] (U : Subring R') [IsInvariantSubring M U] : ↥U →+*[M] R'

The canonical inclusion from an invariant subring.

Equations

• IsInvariantSubring.subtypeHom M U = { toFun := (↑↑U.subtype).toFun, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem IsInvariantSubring.coe_subtypeHom (M : Type u_1) [Monoid M] {R' : Type u_2} [Ring R'] [MulSemiringAction M R'] (U : Subring R') [IsInvariantSubring M U] : ⇑(IsInvariantSubring.subtypeHom M U) = Subtype.val

@[simp]

theorem IsInvariantSubring.coe_subtypeHom' (M : Type u_1) [Monoid M] {R' : Type u_2} [Ring R'] [MulSemiringAction M R'] (U : Subring R') [IsInvariantSubring M U] : (IsInvariantSubring.subtypeHom M U).toRingHom = U.subtype