Finite rings

Content created by Fredrik Bakke, Egbert Rijke and Victor Blanchi.

Created on 2023-05-25.
Last modified on 2024-03-11.

module finite-algebra.finite-rings where
open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.natural-numbers

open import finite-group-theory.finite-abelian-groups
open import finite-group-theory.finite-groups
open import finite-group-theory.finite-monoids

open import foundation.binary-embeddings
open import foundation.binary-equivalences
open import foundation.embeddings
open import foundation.equivalences
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.involutions
open import foundation.propositions
open import foundation.sets
open import foundation.unital-binary-operations
open import foundation.universe-levels

open import group-theory.abelian-groups
open import group-theory.commutative-monoids
open import group-theory.groups
open import group-theory.monoids
open import group-theory.semigroups

open import lists.concatenation-lists
open import lists.lists

open import ring-theory.rings
open import ring-theory.semirings

open import univalent-combinatorics.cartesian-product-types
open import univalent-combinatorics.dependent-function-types
open import univalent-combinatorics.dependent-pair-types
open import univalent-combinatorics.equality-finite-types
open import univalent-combinatorics.finite-types


A finite ring is a ring where the underlying type is finite.


Finite Rings

has-mul-Ab-𝔽 : {l1 : Level} (A : Ab-𝔽 l1)  UU l1
has-mul-Ab-𝔽 A = has-mul-Ab (ab-Ab-𝔽 A)

Ring-𝔽 : (l1 : Level)  UU (lsuc l1)
Ring-𝔽 l1 = Σ (Ab-𝔽 l1)  A  has-mul-Ab-𝔽 A)

finite-ring-is-finite-Ring :
  {l : Level}  (R : Ring l)  is-finite (type-Ring R)  Ring-𝔽 l
pr1 (finite-ring-is-finite-Ring R f) =
  finite-abelian-group-is-finite-Ab (ab-Ring R) f
pr2 (finite-ring-is-finite-Ring R f) = pr2 R

module _
  {l : Level} (R : Ring-𝔽 l)

  finite-ab-Ring-𝔽 : Ab-𝔽 l
  finite-ab-Ring-𝔽 = pr1 R

  ab-Ring-𝔽 : Ab l
  ab-Ring-𝔽 = ab-Ab-𝔽 finite-ab-Ring-𝔽

  ring-Ring-𝔽 : Ring l
  pr1 ring-Ring-𝔽 = ab-Ring-𝔽
  pr2 ring-Ring-𝔽 = pr2 R

  finite-type-Ring-𝔽 : 𝔽 l
  finite-type-Ring-𝔽 = finite-type-Ab-𝔽 finite-ab-Ring-𝔽

  type-Ring-𝔽 : UU l
  type-Ring-𝔽 = type-Ab-𝔽 finite-ab-Ring-𝔽

  is-finite-type-Ring-𝔽 : is-finite type-Ring-𝔽
  is-finite-type-Ring-𝔽 = is-finite-type-Ab-𝔽 finite-ab-Ring-𝔽

  finite-group-Ring-𝔽 : Group-𝔽 l
  finite-group-Ring-𝔽 = finite-group-Ab-𝔽 finite-ab-Ring-𝔽

  group-Ring-𝔽 : Group l
  group-Ring-𝔽 = group-Ab ab-Ring-𝔽

  additive-commutative-monoid-Ring-𝔽 : Commutative-Monoid l
  additive-commutative-monoid-Ring-𝔽 =
    commutative-monoid-Ab ab-Ring-𝔽

  additive-monoid-Ring-𝔽 : Monoid l
  additive-monoid-Ring-𝔽 = monoid-Ab ab-Ring-𝔽

  additive-semigroup-Ring-𝔽 : Semigroup l
  additive-semigroup-Ring-𝔽 = semigroup-Ab ab-Ring-𝔽

  set-Ring-𝔽 : Set l
  set-Ring-𝔽 = set-Ab ab-Ring-𝔽

  is-set-type-Ring-𝔽 : is-set type-Ring-𝔽
  is-set-type-Ring-𝔽 = is-set-type-Ab ab-Ring-𝔽

Addition in a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  has-associative-add-Ring-𝔽 : has-associative-mul-Set (set-Ring-𝔽 R)
  has-associative-add-Ring-𝔽 = has-associative-add-Ring (ring-Ring-𝔽 R)

  add-Ring-𝔽 : type-Ring-𝔽 R  type-Ring-𝔽 R  type-Ring-𝔽 R
  add-Ring-𝔽 = add-Ring (ring-Ring-𝔽 R)

  add-Ring-𝔽' : type-Ring-𝔽 R  type-Ring-𝔽 R  type-Ring-𝔽 R
  add-Ring-𝔽' = add-Ring' (ring-Ring-𝔽 R)

  ap-add-Ring-𝔽 :
    {x y x' y' : type-Ring-𝔽 R} 
    Id x x'  Id y y'  Id (add-Ring-𝔽 x y) (add-Ring-𝔽 x' y')
  ap-add-Ring-𝔽 = ap-add-Ring (ring-Ring-𝔽 R)

  associative-add-Ring-𝔽 :
    (x y z : type-Ring-𝔽 R) 
    Id (add-Ring-𝔽 (add-Ring-𝔽 x y) z) (add-Ring-𝔽 x (add-Ring-𝔽 y z))
  associative-add-Ring-𝔽 = associative-add-Ring (ring-Ring-𝔽 R)

  is-group-additive-semigroup-Ring-𝔽 :
    is-group-Semigroup (additive-semigroup-Ring-𝔽 R)
  is-group-additive-semigroup-Ring-𝔽 =
    is-group-additive-semigroup-Ring (ring-Ring-𝔽 R)

  commutative-add-Ring-𝔽 :
    (x y : type-Ring-𝔽 R)  Id (add-Ring-𝔽 x y) (add-Ring-𝔽 y x)
  commutative-add-Ring-𝔽 = commutative-add-Ring (ring-Ring-𝔽 R)

  interchange-add-add-Ring-𝔽 :
    (x y x' y' : type-Ring-𝔽 R) 
    ( add-Ring-𝔽 (add-Ring-𝔽 x y) (add-Ring-𝔽 x' y')) 
    ( add-Ring-𝔽 (add-Ring-𝔽 x x') (add-Ring-𝔽 y y'))
  interchange-add-add-Ring-𝔽 =
    interchange-add-add-Ring (ring-Ring-𝔽 R)

  right-swap-add-Ring-𝔽 :
    (x y z : type-Ring-𝔽 R) 
    add-Ring-𝔽 (add-Ring-𝔽 x y) z  add-Ring-𝔽 (add-Ring-𝔽 x z) y
  right-swap-add-Ring-𝔽 = right-swap-add-Ring (ring-Ring-𝔽 R)

  left-swap-add-Ring-𝔽 :
    (x y z : type-Ring-𝔽 R) 
    add-Ring-𝔽 x (add-Ring-𝔽 y z)  add-Ring-𝔽 y (add-Ring-𝔽 x z)
  left-swap-add-Ring-𝔽 = left-swap-add-Ring (ring-Ring-𝔽 R)

  is-equiv-add-Ring-𝔽 : (x : type-Ring-𝔽 R)  is-equiv (add-Ring-𝔽 x)
  is-equiv-add-Ring-𝔽 = is-equiv-add-Ring (ring-Ring-𝔽 R)

  is-equiv-add-Ring-𝔽' : (x : type-Ring-𝔽 R)  is-equiv (add-Ring-𝔽' x)
  is-equiv-add-Ring-𝔽' = is-equiv-add-Ring' (ring-Ring-𝔽 R)

  is-binary-equiv-add-Ring-𝔽 : is-binary-equiv add-Ring-𝔽
  is-binary-equiv-add-Ring-𝔽 = is-binary-equiv-add-Ring (ring-Ring-𝔽 R)

  is-binary-emb-add-Ring-𝔽 : is-binary-emb add-Ring-𝔽
  is-binary-emb-add-Ring-𝔽 = is-binary-emb-add-Ring (ring-Ring-𝔽 R)

  is-emb-add-Ring-𝔽 : (x : type-Ring-𝔽 R)  is-emb (add-Ring-𝔽 x)
  is-emb-add-Ring-𝔽 = is-emb-add-Ring (ring-Ring-𝔽 R)

  is-emb-add-Ring-𝔽' : (x : type-Ring-𝔽 R)  is-emb (add-Ring-𝔽' x)
  is-emb-add-Ring-𝔽' = is-emb-add-Ring' (ring-Ring-𝔽 R)

  is-injective-add-Ring-𝔽 : (x : type-Ring-𝔽 R)  is-injective (add-Ring-𝔽 x)
  is-injective-add-Ring-𝔽 = is-injective-add-Ring (ring-Ring-𝔽 R)

  is-injective-add-Ring-𝔽' : (x : type-Ring-𝔽 R)  is-injective (add-Ring-𝔽' x)
  is-injective-add-Ring-𝔽' = is-injective-add-Ring' (ring-Ring-𝔽 R)

The zero element of a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  has-zero-Ring-𝔽 : is-unital (add-Ring-𝔽 R)
  has-zero-Ring-𝔽 = has-zero-Ring (ring-Ring-𝔽 R)

  zero-Ring-𝔽 : type-Ring-𝔽 R
  zero-Ring-𝔽 = zero-Ring (ring-Ring-𝔽 R)

  is-zero-Ring-𝔽 : type-Ring-𝔽 R  UU l
  is-zero-Ring-𝔽 = is-zero-Ring (ring-Ring-𝔽 R)

  is-nonzero-Ring-𝔽 : type-Ring-𝔽 R  UU l
  is-nonzero-Ring-𝔽 = is-nonzero-Ring (ring-Ring-𝔽 R)

  is-zero-finite-ring-Prop : type-Ring-𝔽 R  Prop l
  is-zero-finite-ring-Prop = is-zero-ring-Prop (ring-Ring-𝔽 R)

  is-nonzero-finite-ring-Prop : type-Ring-𝔽 R  Prop l
  is-nonzero-finite-ring-Prop = is-nonzero-ring-Prop (ring-Ring-𝔽 R)

  left-unit-law-add-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (add-Ring-𝔽 R zero-Ring-𝔽 x) x
  left-unit-law-add-Ring-𝔽 = left-unit-law-add-Ring (ring-Ring-𝔽 R)

  right-unit-law-add-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (add-Ring-𝔽 R x zero-Ring-𝔽) x
  right-unit-law-add-Ring-𝔽 = right-unit-law-add-Ring (ring-Ring-𝔽 R)

Additive inverses in a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  has-negatives-Ring-𝔽 :
      ( additive-semigroup-Ring-𝔽 R)
      ( has-zero-Ring-𝔽 R)
  has-negatives-Ring-𝔽 = has-negatives-Ring (ring-Ring-𝔽 R)

  neg-Ring-𝔽 : type-Ring-𝔽 R  type-Ring-𝔽 R
  neg-Ring-𝔽 = neg-Ring (ring-Ring-𝔽 R)

  left-inverse-law-add-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (add-Ring-𝔽 R (neg-Ring-𝔽 x) x) (zero-Ring-𝔽 R)
  left-inverse-law-add-Ring-𝔽 = left-inverse-law-add-Ring (ring-Ring-𝔽 R)

  right-inverse-law-add-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (add-Ring-𝔽 R x (neg-Ring-𝔽 x)) (zero-Ring-𝔽 R)
  right-inverse-law-add-Ring-𝔽 = right-inverse-law-add-Ring (ring-Ring-𝔽 R)

  neg-neg-Ring-𝔽 : (x : type-Ring-𝔽 R)  neg-Ring-𝔽 (neg-Ring-𝔽 x)  x
  neg-neg-Ring-𝔽 = neg-neg-Ring (ring-Ring-𝔽 R)

  distributive-neg-add-Ring-𝔽 :
    (x y : type-Ring-𝔽 R) 
    neg-Ring-𝔽 (add-Ring-𝔽 R x y)  add-Ring-𝔽 R (neg-Ring-𝔽 x) (neg-Ring-𝔽 y)
  distributive-neg-add-Ring-𝔽 = distributive-neg-add-Ring (ring-Ring-𝔽 R)

Multiplication in a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  has-associative-mul-Ring-𝔽 : has-associative-mul-Set (set-Ring-𝔽 R)
  has-associative-mul-Ring-𝔽 = has-associative-mul-Ring (ring-Ring-𝔽 R)

  mul-Ring-𝔽 : type-Ring-𝔽 R  type-Ring-𝔽 R  type-Ring-𝔽 R
  mul-Ring-𝔽 = mul-Ring (ring-Ring-𝔽 R)

  mul-Ring-𝔽' : type-Ring-𝔽 R  type-Ring-𝔽 R  type-Ring-𝔽 R
  mul-Ring-𝔽' = mul-Ring' (ring-Ring-𝔽 R)

  ap-mul-Ring-𝔽 :
    {x x' y y' : type-Ring-𝔽 R} (p : Id x x') (q : Id y y') 
    Id (mul-Ring-𝔽 x y) (mul-Ring-𝔽 x' y')
  ap-mul-Ring-𝔽 = ap-mul-Ring (ring-Ring-𝔽 R)

  associative-mul-Ring-𝔽 :
    (x y z : type-Ring-𝔽 R) 
    Id (mul-Ring-𝔽 (mul-Ring-𝔽 x y) z) (mul-Ring-𝔽 x (mul-Ring-𝔽 y z))
  associative-mul-Ring-𝔽 = associative-mul-Ring (ring-Ring-𝔽 R)

  multiplicative-semigroup-Ring-𝔽 : Semigroup l
  multiplicative-semigroup-Ring-𝔽 =
    multiplicative-semigroup-Ring (ring-Ring-𝔽 R)

  left-distributive-mul-add-Ring-𝔽 :
    (x y z : type-Ring-𝔽 R) 
    mul-Ring-𝔽 x (add-Ring-𝔽 R y z) 
    add-Ring-𝔽 R (mul-Ring-𝔽 x y) (mul-Ring-𝔽 x z)
  left-distributive-mul-add-Ring-𝔽 =
    left-distributive-mul-add-Ring (ring-Ring-𝔽 R)

  right-distributive-mul-add-Ring-𝔽 :
    (x y z : type-Ring-𝔽 R) 
    mul-Ring-𝔽 (add-Ring-𝔽 R x y) z 
    add-Ring-𝔽 R (mul-Ring-𝔽 x z) (mul-Ring-𝔽 y z)
  right-distributive-mul-add-Ring-𝔽 =
    right-distributive-mul-add-Ring (ring-Ring-𝔽 R)

Multiplicative units in a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  is-unital-Ring-𝔽 : is-unital (mul-Ring-𝔽 R)
  is-unital-Ring-𝔽 = is-unital-Ring (ring-Ring-𝔽 R)

  multiplicative-monoid-Ring-𝔽 : Monoid l
  multiplicative-monoid-Ring-𝔽 = multiplicative-monoid-Ring (ring-Ring-𝔽 R)

  one-Ring-𝔽 : type-Ring-𝔽 R
  one-Ring-𝔽 = one-Ring (ring-Ring-𝔽 R)

  left-unit-law-mul-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (mul-Ring-𝔽 R one-Ring-𝔽 x) x
  left-unit-law-mul-Ring-𝔽 = left-unit-law-mul-Ring (ring-Ring-𝔽 R)

  right-unit-law-mul-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (mul-Ring-𝔽 R x one-Ring-𝔽) x
  right-unit-law-mul-Ring-𝔽 = right-unit-law-mul-Ring (ring-Ring-𝔽 R)

The zero laws for multiplication of a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  left-zero-law-mul-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (mul-Ring-𝔽 R (zero-Ring-𝔽 R) x) (zero-Ring-𝔽 R)
  left-zero-law-mul-Ring-𝔽 =
    left-zero-law-mul-Ring (ring-Ring-𝔽 R)

  right-zero-law-mul-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  Id (mul-Ring-𝔽 R x (zero-Ring-𝔽 R)) (zero-Ring-𝔽 R)
  right-zero-law-mul-Ring-𝔽 =
    right-zero-law-mul-Ring (ring-Ring-𝔽 R)

Rings are semirings

module _
  {l : Level} (R : Ring-𝔽 l)

  has-mul-Ring-𝔽 :
    has-mul-Commutative-Monoid (additive-commutative-monoid-Ring-𝔽 R)
  has-mul-Ring-𝔽 = has-mul-Ring (ring-Ring-𝔽 R)

  zero-laws-mul-Ring-𝔽 :
      ( additive-commutative-monoid-Ring-𝔽 R)
      ( has-mul-Ring-𝔽)
  zero-laws-mul-Ring-𝔽 = zero-laws-mul-Ring (ring-Ring-𝔽 R)

  semiring-Ring-𝔽 : Semiring l
  semiring-Ring-𝔽 = semiring-Ring (ring-Ring-𝔽 R)

Computing multiplication with minus one in a ring

module _
  {l : Level} (R : Ring-𝔽 l)

  neg-one-Ring-𝔽 : type-Ring-𝔽 R
  neg-one-Ring-𝔽 = neg-one-Ring (ring-Ring-𝔽 R)

  mul-neg-one-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  mul-Ring-𝔽 R neg-one-Ring-𝔽 x  neg-Ring-𝔽 R x
  mul-neg-one-Ring-𝔽 =
    mul-neg-one-Ring (ring-Ring-𝔽 R)

  mul-neg-one-Ring-𝔽' :
    (x : type-Ring-𝔽 R)  mul-Ring-𝔽 R x neg-one-Ring-𝔽  neg-Ring-𝔽 R x
  mul-neg-one-Ring-𝔽' =
    mul-neg-one-Ring' (ring-Ring-𝔽 R)

  is-involution-mul-neg-one-Ring-𝔽 :
    is-involution (mul-Ring-𝔽 R neg-one-Ring-𝔽)
  is-involution-mul-neg-one-Ring-𝔽 =
    is-involution-mul-neg-one-Ring (ring-Ring-𝔽 R)

  is-involution-mul-neg-one-Ring-𝔽' :
    is-involution (mul-Ring-𝔽' R neg-one-Ring-𝔽)
  is-involution-mul-neg-one-Ring-𝔽' =
    is-involution-mul-neg-one-Ring' (ring-Ring-𝔽 R)

Left and right negative laws for multiplication

module _
  {l : Level} (R : Ring-𝔽 l)

  left-negative-law-mul-Ring-𝔽 :
    (x y : type-Ring-𝔽 R) 
    mul-Ring-𝔽 R (neg-Ring-𝔽 R x) y  neg-Ring-𝔽 R (mul-Ring-𝔽 R x y)
  left-negative-law-mul-Ring-𝔽 =
    left-negative-law-mul-Ring (ring-Ring-𝔽 R)

  right-negative-law-mul-Ring-𝔽 :
    (x y : type-Ring-𝔽 R) 
    mul-Ring-𝔽 R x (neg-Ring-𝔽 R y)  neg-Ring-𝔽 R (mul-Ring-𝔽 R x y)
  right-negative-law-mul-Ring-𝔽 =
    right-negative-law-mul-Ring (ring-Ring-𝔽 R)

  mul-neg-Ring-𝔽 :
    (x y : type-Ring-𝔽 R) 
    mul-Ring-𝔽 R (neg-Ring-𝔽 R x) (neg-Ring-𝔽 R y)  mul-Ring-𝔽 R x y
  mul-neg-Ring-𝔽 =
    mul-neg-Ring (ring-Ring-𝔽 R)

Scalar multiplication of ring elements by a natural number

module _
  {l : Level} (R : Ring-𝔽 l)

  mul-nat-scalar-Ring-𝔽 :   type-Ring-𝔽 R  type-Ring-𝔽 R
  mul-nat-scalar-Ring-𝔽 = mul-nat-scalar-Ring (ring-Ring-𝔽 R)

  ap-mul-nat-scalar-Ring-𝔽 :
    {m n : } {x y : type-Ring-𝔽 R} 
    (m  n)  (x  y)  mul-nat-scalar-Ring-𝔽 m x  mul-nat-scalar-Ring-𝔽 n y
  ap-mul-nat-scalar-Ring-𝔽 = ap-mul-nat-scalar-Ring (ring-Ring-𝔽 R)

  left-zero-law-mul-nat-scalar-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  mul-nat-scalar-Ring-𝔽 0 x  zero-Ring-𝔽 R
  left-zero-law-mul-nat-scalar-Ring-𝔽 =
    left-zero-law-mul-nat-scalar-Ring (ring-Ring-𝔽 R)

  right-zero-law-mul-nat-scalar-Ring-𝔽 :
    (n : )  mul-nat-scalar-Ring-𝔽 n (zero-Ring-𝔽 R)  zero-Ring-𝔽 R
  right-zero-law-mul-nat-scalar-Ring-𝔽 =
    right-zero-law-mul-nat-scalar-Ring (ring-Ring-𝔽 R)

  left-unit-law-mul-nat-scalar-Ring-𝔽 :
    (x : type-Ring-𝔽 R)  mul-nat-scalar-Ring-𝔽 1 x  x
  left-unit-law-mul-nat-scalar-Ring-𝔽 =
    left-unit-law-mul-nat-scalar-Ring (ring-Ring-𝔽 R)

  left-nat-scalar-law-mul-Ring-𝔽 :
    (n : ) (x y : type-Ring-𝔽 R) 
    mul-Ring-𝔽 R (mul-nat-scalar-Ring-𝔽 n x) y 
    mul-nat-scalar-Ring-𝔽 n (mul-Ring-𝔽 R x y)
  left-nat-scalar-law-mul-Ring-𝔽 =
    left-nat-scalar-law-mul-Ring (ring-Ring-𝔽 R)

  right-nat-scalar-law-mul-Ring-𝔽 :
    (n : ) (x y : type-Ring-𝔽 R) 
    mul-Ring-𝔽 R x (mul-nat-scalar-Ring-𝔽 n y) 
    mul-nat-scalar-Ring-𝔽 n (mul-Ring-𝔽 R x y)
  right-nat-scalar-law-mul-Ring-𝔽 =
    right-nat-scalar-law-mul-Ring (ring-Ring-𝔽 R)

  left-distributive-mul-nat-scalar-add-Ring-𝔽 :
    (n : ) (x y : type-Ring-𝔽 R) 
    mul-nat-scalar-Ring-𝔽 n (add-Ring-𝔽 R x y) 
    add-Ring-𝔽 R (mul-nat-scalar-Ring-𝔽 n x) (mul-nat-scalar-Ring-𝔽 n y)
  left-distributive-mul-nat-scalar-add-Ring-𝔽 =
    left-distributive-mul-nat-scalar-add-Ring (ring-Ring-𝔽 R)

  right-distributive-mul-nat-scalar-add-Ring-𝔽 :
    (m n : ) (x : type-Ring-𝔽 R) 
    mul-nat-scalar-Ring-𝔽 (m +ℕ n) x 
    add-Ring-𝔽 R (mul-nat-scalar-Ring-𝔽 m x) (mul-nat-scalar-Ring-𝔽 n x)
  right-distributive-mul-nat-scalar-add-Ring-𝔽 =
    right-distributive-mul-nat-scalar-add-Ring (ring-Ring-𝔽 R)

Addition of a list of elements in an abelian group

module _
  {l : Level} (R : Ring-𝔽 l)

  add-list-Ring-𝔽 : list (type-Ring-𝔽 R)  type-Ring-𝔽 R
  add-list-Ring-𝔽 = add-list-Ring (ring-Ring-𝔽 R)

  preserves-concat-add-list-Ring-𝔽 :
    (l1 l2 : list (type-Ring-𝔽 R)) 
      ( add-list-Ring-𝔽 (concat-list l1 l2))
      ( add-Ring-𝔽 R (add-list-Ring-𝔽 l1) (add-list-Ring-𝔽 l2))
  preserves-concat-add-list-Ring-𝔽 =
    preserves-concat-add-list-Ring (ring-Ring-𝔽 R)


There is a finite number of ways to equip a finite type with the structure of a ring

module _
  {l : Level}
  (X : 𝔽 l)

  structure-ring-𝔽 : UU l
  structure-ring-𝔽 =
    Σ ( structure-abelian-group-𝔽 X)
      ( λ m  has-mul-Ab-𝔽 (finite-abelian-group-structure-abelian-group-𝔽 X m))

  finite-ring-structure-ring-𝔽 :
    structure-ring-𝔽  Ring-𝔽 l
  pr1 (finite-ring-structure-ring-𝔽 (m , c)) =
    finite-abelian-group-structure-abelian-group-𝔽 X m
  pr2 (finite-ring-structure-ring-𝔽 (m , c)) = c

  is-finite-structure-ring-𝔽 :
    is-finite structure-ring-𝔽
  is-finite-structure-ring-𝔽 =
      ( is-finite-structure-abelian-group-𝔽 X)
      ( λ a 
          ( is-finite-Σ
            ( is-finite-Π
              ( is-finite-type-𝔽 X)
              ( λ _ 
                  ( is-finite-type-𝔽 X)
                  ( λ _  is-finite-type-𝔽 X)))
            ( λ m 
                ( is-finite-type-𝔽 X)
                ( λ x 
                    ( is-finite-type-𝔽 X)
                    ( λ y 
                        ( is-finite-type-𝔽 X)
                        ( λ z  is-finite-eq-𝔽 X)))))
          ( λ a 
              ( is-finite-is-unital-Semigroup-𝔽 (X , a))
              ( is-finite-product
                ( is-finite-Π
                  ( is-finite-type-𝔽 X)
                  ( λ _ 
                      ( is-finite-type-𝔽 X)
                      ( λ _ 
                          ( is-finite-type-𝔽 X)
                          ( λ _  is-finite-eq-𝔽 X))))
                ( is-finite-Π
                  ( is-finite-type-𝔽 X)
                  ( λ _ 
                      ( is-finite-type-𝔽 X)
                      ( λ _ 
                          ( is-finite-type-𝔽 X)
                          ( λ _  is-finite-eq-𝔽 X)))))))

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