open import Algebra.Group.Cat.Base
open import Algebra.Semigroup
open import Algebra.Group.Ab hiding (ℤ)
open import Algebra.Prelude
open import Algebra.Monoid
open import Algebra.Group

open import Cat.Instances.Delooping
open import Cat.Abelian.Base

open import Data.Int.Properties
open import Data.Int.Base

import Cat.Reasoning

module Algebra.Ring where

Rings🔗

The ring is one of the basic objects of study in algebra, which abstracts the best bits of the common algebraic structures: The integers $\mathbb{Z}\text{,}$ the rationals $\mathbb{Q}\text{,}$ the reals $\mathbb{R}\text{,}$ and the complex numbers $\mathbb{C}$ are all rings, as are the collections of polynomials with coefficients in any of those. Less familiar examples of rings include square matrices (with values in a ring) and the integral cohomology ring of a topological space: that these are so far from being “number-like” indicates the incredible generality of rings.

A ring is an abelian group $R$ (which we call the additive group of $R\text{),}$ together with the data of a monoid on $R$ (the multiplicative monoid), where the multiplication of the monoid distributes over the addition. We’ll see why this compatibility condition is required afterwards. Check out what it means for a triple $(1,\ast ,+)$ to be a ring structure on a type:

record is-ring {ℓ} {R : Type ℓ} (1r : R) (_*_ _+_ : R → R → R) : Type ℓ where
  no-eta-equality
  field
    *-monoid : is-monoid 1r _*_
    +-group  : is-abelian-group _+_
    *-distribl : ∀ {x y z} → x * (y + z) ≡ (x * y) + (x * z)
    *-distribr : ∀ {x y z} → (y + z) * x ≡ (y * x) + (z * x)

  open is-monoid *-monoid
    renaming ( idl to *-idl
             ; idr to *-idr
             ; associative to *-associative
             )
    hiding (has-is-set ; magma-hlevel ; underlying-set)
    public

  open is-abelian-group +-group
    renaming ( _—_ to _-_
             ; inverse to -_
             ; 1g to 0r
             ; inversel to +-invl
             ; inverser to +-invr
             ; associative to +-associative
             ; idl to +-idl
             ; idr to +-idr
             ; commutes to +-commutes
             )
    public

  additive-group : Group ℓ
  ∣ additive-group .fst ∣                    = R
  additive-group .fst .is-tr                 = is-abelian-group.has-is-set +-group
  additive-group .snd .Group-on._⋆_          = _+_
  additive-group .snd .Group-on.has-is-group = is-abelian-group.has-is-group +-group

  group : Abelian-group ℓ
  ∣ group .fst ∣                         = R
  group .fst .is-tr                      = is-abelian-group.has-is-set +-group
  group .snd .Abelian-group-on._*_       = _+_
  group .snd .Abelian-group-on.has-is-ab = +-group

  Ringoid : Ab-category (B record { _⋆_ = _*_ ; has-is-monoid = *-monoid })
  Ringoid .Ab-category.Abelian-group-on-hom _ _ = record { has-is-ab = +-group }
  Ringoid .Ab-category.∘-linear-l f g h = sym *-distribr
  Ringoid .Ab-category.∘-linear-r f g h = sym *-distribl

  private
    module ringoid = Ab-category Ringoid
      using ( ∘-zero-l ; ∘-zero-r ; neg-∘-l ; neg-∘-r ; ∘-minus-l ; ∘-minus-r )

  open ringoid renaming
      ( ∘-zero-l to *-zerol
      ; ∘-zero-r to *-zeror
      ; neg-∘-l to neg-*-l
      ; neg-∘-r to neg-*-r
      ; ∘-minus-l to *-minus-l
      ; ∘-minus-r to *-minus-r
      )
    public

  module m = Cat.Reasoning (B record { _⋆_ = _*_ ; has-is-monoid = *-monoid })
  module a = Abelian-group-on record { has-is-ab = +-group }

record Ring-on {ℓ} (R : Type ℓ) : Type ℓ where
  field
    1r : R
    _*_ _+_ : R → R → R
    has-is-ring : is-ring 1r _*_ _+_

  open is-ring has-is-ring public
  infixl  _*_
  infixl  _+_

instance
  H-Level-is-ring
    : ∀ {ℓ} {R : Type ℓ} {1r : R} {_*_ _+_ : R → R → R} {n}
    → H-Level (is-ring 1r _*_ _+_) (suc n)
  H-Level-is-ring {1r = 1r} {_*_} {_+_} =
    prop-instance {T = is-ring 1r _*_ _+_} $ λ where
      x y i .*-monoid   → hlevel  (x .*-monoid) (y .*-monoid) i
      x y i .+-group    → hlevel  (x .+-group) (y .+-group) i
      x y i .*-distribl → x .+-group .is-abelian-group.has-is-set _ _ (x .*-distribl) (y .*-distribl) i
      x y i .*-distribr → x .+-group .is-abelian-group.has-is-set _ _ (x .*-distribr) (y .*-distribr) i
    where open is-ring

There is a natural notion of ring homomorphism, which we get by smashing together that of a monoid homomorphism (for the multiplicative part) and of group homomorphism; Every map of rings has an underlying map of groups which preserves the addition operation, and it must also preserve the multiplication. This encodes the view of a ring as an “abelian group with a monoid structure”.

record is-ring-hom
  {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (R : Ring-on A) (S : Ring-on B)
  (f : A → B)
  : Type (ℓ ⊔ ℓ') where
  private
    module A = Ring-on R
    module B = Ring-on S

  field
    pres-id : f A.1r ≡ B.1r
    pres-+  : ∀ x y → f (x A.+ y) ≡ f x B.+ f y
    pres-*  : ∀ x y → f (x A.* y) ≡ f x B.* f y

  ring-hom→group-hom : is-group-hom (A.additive-group .snd) (B.additive-group .snd) f
  ring-hom→group-hom = record { pres-⋆ = pres-+ }

  module gh = is-group-hom ring-hom→group-hom renaming (pres-id to pres-0 ; pres-inv to pres-neg)
  open gh using (pres-0 ; pres-neg ; pres-diff) public

private unquoteDecl eqv = declare-record-iso eqv (quote is-ring-hom)

module _ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} {R : Ring-on A} {S : Ring-on B} where
  open Ring-on R using (magma-hlevel)
  open Ring-on S using (magma-hlevel)

  instance abstract
    H-Level-ring-hom : ∀ {f n} → H-Level (is-ring-hom R S f) (suc n)
    H-Level-ring-hom = prop-instance λ x y → Iso→is-hlevel  eqv (hlevel ) x y

open is-ring-hom

It follows, by standard equational nonsense, that rings and ring homomorphisms form a precategory — for instance, we have $f(g(1_R))=f(1_S)=1_T\text{.}$

Ring-structure : ∀ ℓ → Thin-structure ℓ Ring-on
Ring-structure ℓ .is-hom f x y = el! (is-ring-hom x y f)
Ring-structure ℓ .id-is-hom .pres-id = refl
Ring-structure ℓ .id-is-hom .pres-+ x y = refl
Ring-structure ℓ .id-is-hom .pres-* x y = refl
Ring-structure ℓ .∘-is-hom f g α β .pres-id = ap f (β .pres-id) ∙ α .pres-id
Ring-structure ℓ .∘-is-hom f g α β .pres-+ x y = ap f (β .pres-+ x y) ∙ α .pres-+ _ _
Ring-structure ℓ .∘-is-hom f g α β .pres-* x y = ap f (β .pres-* x y) ∙ α .pres-* _ _
Ring-structure ℓ .id-hom-unique α β i .Ring-on.1r = α .pres-id i
Ring-structure ℓ .id-hom-unique α β i .Ring-on._*_ x y = α .pres-* x y i
Ring-structure ℓ .id-hom-unique α β i .Ring-on._+_ x y = α .pres-+ x y i
Ring-structure ℓ .id-hom-unique {s = s} {t} α β i .Ring-on.has-is-ring =
  is-prop→pathp
    (λ i → hlevel {T = is-ring (α .pres-id i)
      (λ x y → α .pres-* x y i) (λ x y → α .pres-+ x y i)} )
    (s .Ring-on.has-is-ring) (t .Ring-on.has-is-ring) i

Rings : ∀ ℓ → Precategory (lsuc ℓ) ℓ
Rings _ = Structured-objects (Ring-structure _)
module Rings {ℓ} = Cat.Reasoning (Rings ℓ)

Ring : ∀ ℓ → Type (lsuc ℓ)
Ring ℓ = Rings.Ob

In components🔗

We give a more elementary description of rings, which is suitable for constructing values of the record type Ring above. This re-expresses the data included in the definition of a ring with the least amount of redundancy possible, in the most direct terms possible: A ring is a set, equipped with two binary operations $\ast$ and $+\text{,}$ such that $\ast$ distributes over $+$ on either side; $+$ is an abelian group; and $\ast$ is a monoid.

record make-ring {ℓ} (R : Type ℓ) : Type ℓ where
  no-eta-equality
  field
    ring-is-set : is-set R

    -- R is an abelian group:
    0R      : R
    _+_     : R → R → R
    -_      : R → R
    +-idl   : ∀ x → 0R + x ≡ x
    +-invr  : ∀ x → x + (- x) ≡ 0R
    +-assoc : ∀ x y z → x + (y + z) ≡ (x + y) + z
    +-comm  : ∀ x y → x + y ≡ y + x

    -- R is a commutative monoid:
    1R      : R
    _*_     : R → R → R
    *-idl   : ∀ x → 1R * x ≡ x
    *-idr   : ∀ x → x * 1R ≡ x
    *-assoc : ∀ x y z → x * (y * z) ≡ (x * y) * z

    -- Multiplication is bilinear:
    *-distribl : ∀ x y z → x * (y + z) ≡ (x * y) + (x * z)
    *-distribr : ∀ x y z → (y + z) * x ≡ (y * x) + (z * x)

  to-ring-on : Ring-on R
  to-ring-on = ring where
    open is-ring hiding (-_ ; +-invr ; +-invl ; *-distribl ; *-distribr ; *-idl ; *-idr ; +-idl ; +-idr)

    -- All in copatterns to prevent the unfolding from exploding on you
    ring : Ring-on R
    ring .Ring-on.1r = 1R
    ring .Ring-on._*_ = _*_
    ring .Ring-on._+_ = _+_
    ring .Ring-on.has-is-ring .*-monoid .has-is-semigroup .is-semigroup.has-is-magma = record { has-is-set = ring-is-set }
    ring .Ring-on.has-is-ring .*-monoid .has-is-semigroup .is-semigroup.associative = *-assoc _ _ _
    ring .Ring-on.has-is-ring .*-monoid .idl = *-idl _
    ring .Ring-on.has-is-ring .*-monoid .idr = *-idr _
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.unit = 0R
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.has-is-monoid .has-is-semigroup .has-is-magma = record { has-is-set = ring-is-set }
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.has-is-monoid .has-is-semigroup .associative = +-assoc _ _ _
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.has-is-monoid .idl = +-idl _
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.has-is-monoid .idr = +-comm _ _ ∙ +-idl _
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.inverse = -_
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.inversel = +-comm _ _ ∙ +-invr _
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.has-is-group .is-group.inverser = +-invr _
    ring .Ring-on.has-is-ring .+-group .is-abelian-group.commutes = +-comm _ _
    ring .Ring-on.has-is-ring .is-ring.*-distribl = *-distribl _ _ _
    ring .Ring-on.has-is-ring .is-ring.*-distribr = *-distribr _ _ _

  to-ring : Ring ℓ
  to-ring .fst = el R ring-is-set
  to-ring .snd = to-ring-on

open make-ring using (to-ring ; to-ring-on) public

This data is missing (by design, actually!) one condition which we would expect: $0\ne 1\text{.}$ We exploit this to give our first example of a ring, the zero ring, which has carrier set the unit — the type with one object.

Despite the name, the zero ring is not the zero object in the category of rings: it is the terminal object. In the category of rings, the initial object is the ring $\mathbb{Z}\text{,}$ which is very far (infinitely far!) from having a single element. It’s called the “zero ring” because it has one element $x\text{,}$ which must be the additive identity, hence we call it $0\text{.}$ But it’s also the multiplicative identity, so we might also call the ring $\{\ast \}$ the One Ring, which would be objectively cooler.

Zero-ring : Ring lzero
Zero-ring = to-ring {R = ⊤} λ where
  .make-ring.ring-is-set _ _ _ _ _ _ → tt
  .make-ring.0R                      → tt
  .make-ring._+_ _ _                 → tt
  .make-ring.-_  _                   → tt
  .make-ring.+-idl  _ _              → tt
  .make-ring.+-invr _ _              → tt
  .make-ring.+-assoc _ _ _ _         → tt
  .make-ring.+-comm _ _ _            → tt
  .make-ring.1R                      → tt
  .make-ring._*_ _ _                 → tt
  .make-ring.*-idl _ _               → tt
  .make-ring.*-idr _ _               → tt
  .make-ring.*-assoc _ _ _ _         → tt
  .make-ring.*-distribl _ _ _ _      → tt
  .make-ring.*-distribr _ _ _ _      → tt

Rings, unlike other categories of algebraic structures (like that of groups or abelian groups), are structured enough to differentiate between the initial and terminal objects. As mentioned above, the initial object is the ring $\mathbb{Z}\text{,}$ and the terminal ring is the zero ring. As for why this happens, consider that, since ring homomorphisms must preserve the unit¹, it is impossible to have a ring homomorphism $h:0\to R$ unless $0=h(0)=h(1)=1$ in $R\text{.}$

ℤ : Ring lzero
ℤ = to-ring {R = Int} λ where
  .make-ring.ring-is-set → hlevel 
  .make-ring.1R         → 
  .make-ring.0R         → 
  .make-ring._+_        → _+ℤ_
  .make-ring.-_         → negℤ
  .make-ring._*_        → _*ℤ_
  .make-ring.+-idl      → +ℤ-zerol
  .make-ring.+-invr     → +ℤ-invr
  .make-ring.+-assoc    → +ℤ-assoc
  .make-ring.+-comm     → +ℤ-commutative
  .make-ring.*-idl      → *ℤ-onel
  .make-ring.*-idr      → *ℤ-oner
  .make-ring.*-assoc    → *ℤ-associative
  .make-ring.*-distribl → *ℤ-distribl
  .make-ring.*-distribr → *ℤ-distribr