Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 4,405 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import algebra.ring.equiv
import group_theory.group_action.group
import ring_theory.subring.basic
/-!
# 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
-/
universes u v
open_locale big_operators
/-- Typeclass for multiplicative actions by monoids on semirings.
This combines `distrib_mul_action` with `mul_distrib_mul_action`. -/
class mul_semiring_action (M : Type u) (R : Type v) [monoid M] [semiring R]
extends distrib_mul_action M R :=
(smul_one : ∀ (g : M), (g • 1 : R) = 1)
(smul_mul : ∀ (g : M) (x y : R), g • (x * y) = (g • x) * (g • y))
section semiring
variables (M G : Type u) [monoid M] [group G]
variables (A R S F : Type v) [add_monoid A] [semiring R] [comm_semiring S] [division_ring F]
-- note we could not use `extends` since these typeclasses are made with `old_structure_cmd`
@[priority 100]
instance mul_semiring_action.to_mul_distrib_mul_action [h : mul_semiring_action M R] :
mul_distrib_mul_action M R :=
{ ..h }
/-- Each element of the monoid defines a semiring homomorphism. -/
@[simps]
def mul_semiring_action.to_ring_hom [mul_semiring_action M R] (x : M) : R →+* R :=
{ .. mul_distrib_mul_action.to_monoid_hom R x,
.. distrib_mul_action.to_add_monoid_hom R x }
theorem to_ring_hom_injective [mul_semiring_action M R] [has_faithful_smul M R] :
function.injective (mul_semiring_action.to_ring_hom M R) :=
λ m₁ m₂ h, eq_of_smul_eq_smul $ λ r, ring_hom.ext_iff.1 h r
/-- Each element of the group defines a semiring isomorphism. -/
@[simps]
def mul_semiring_action.to_ring_equiv [mul_semiring_action G R] (x : G) : R ≃+* R :=
{ .. distrib_mul_action.to_add_equiv R x,
.. mul_semiring_action.to_ring_hom G R x }
section
variables {M G R}
/-- A stronger version of `submonoid.distrib_mul_action`. -/
instance submonoid.mul_semiring_action [mul_semiring_action M R] (H : submonoid M) :
mul_semiring_action H R :=
{ smul := (•),
.. H.mul_distrib_mul_action,
.. H.distrib_mul_action }
/-- A stronger version of `subgroup.distrib_mul_action`. -/
instance subgroup.mul_semiring_action [mul_semiring_action G R] (H : subgroup G) :
mul_semiring_action H R :=
H.to_submonoid.mul_semiring_action
/-- A stronger version of `subsemiring.distrib_mul_action`. -/
instance subsemiring.mul_semiring_action {R'} [semiring R'] [mul_semiring_action R' R]
(H : subsemiring R') :
mul_semiring_action H R :=
H.to_submonoid.mul_semiring_action
/-- A stronger version of `subring.distrib_mul_action`. -/
instance subring.mul_semiring_action {R'} [ring R'] [mul_semiring_action R' R]
(H : subring R') :
mul_semiring_action H R :=
H.to_subsemiring.mul_semiring_action
end
section simp_lemmas
variables {M G A R F}
attribute [simp] smul_one smul_mul' smul_zero smul_add
/-- Note that `smul_inv'` refers to the group case, and `smul_inv` has an additional inverse
on `x`. -/
@[simp] lemma smul_inv'' [mul_semiring_action M F] (x : M) (m : F) : x • m⁻¹ = (x • m)⁻¹ :=
(mul_semiring_action.to_ring_hom M F x).map_inv _
end simp_lemmas
end semiring
section ring
variables (M : Type u) [monoid M] {R : Type v} [ring R] [mul_semiring_action M R]
variables (S : subring R)
open mul_action
/-- A typeclass for subrings invariant under a `mul_semiring_action`. -/
class is_invariant_subring : Prop :=
(smul_mem : ∀ (m : M) {x : R}, x ∈ S → m • x ∈ S)
instance is_invariant_subring.to_mul_semiring_action [is_invariant_subring M S] :
mul_semiring_action M S :=
{ smul := λ m x, ⟨m • x, is_invariant_subring.smul_mem m x.2⟩,
one_smul := λ s, subtype.eq $ one_smul M s,
mul_smul := λ m₁ m₂ s, subtype.eq $ mul_smul m₁ m₂ s,
smul_add := λ m s₁ s₂, subtype.eq $ smul_add m s₁ s₂,
smul_zero := λ m, subtype.eq $ smul_zero m,
smul_one := λ m, subtype.eq $ smul_one m,
smul_mul := λ m s₁ s₂, subtype.eq $ smul_mul' m s₁ s₂ }
end ring
|