{-# OPTIONS -vtc.def.fun:10 #-}
open import Algebra.Group.Cat.Base
open import Algebra.Semigroup
open import Algebra.Group.Ab
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

import Cat.Reasoning

module Algebra.Ring where

RingsπŸ”—

The ring is one of the basic objects of study in algebra, which abstracts the best bits of the common algebraic structures: The integers Z\mathbb{Z}, the rationals Q\mathbb{Q}, the reals R\mathbb{R}, and the complex numbers C\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 β€œnumber-like” indicates the incredible generality of rings.

A ring is an abelian group RR (which we call the additive group of RR), together with the data of a monoid on RR (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,βˆ—,+)(1, *, +) to be a ring structure on a type:

record is-ring {β„“} {R : Type β„“} (1r : R) (_*_ _+_ : R β†’ R β†’ R) : Type β„“ where
  no-eta-equality
  field
    *-monoid : is-monoid 1r _*_
    +-group  : is-abelian-group _+_
    *-distribl : βˆ€ {x y z} β†’ x * (y + z) ≑ (x * y) + (x * z)
    *-distribr : βˆ€ {x y z} β†’ (y + z) * x ≑ (y * x) + (z * x)

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 β€œabelian group with a monoid structure”.

record is-ring-hom
  {β„“ β„“β€²} {A : Type β„“} {B : Type β„“β€²} (R : Ring-on A) (S : Ring-on B)
  (f : A β†’ B)
  : Type (β„“ βŠ” β„“β€²) where
  private
    module A = Ring-on R
    module B = Ring-on S

  field
    pres-id : f A.1r ≑ B.1r
    pres-+  : βˆ€ x y β†’ f (x A.+ y) ≑ f x B.+ f y
    pres-*  : βˆ€ x y β†’ f (x A.* y) ≑ f x B.* f y

It follows, by standard equational nonsense, that rings and ring homomorphisms form a precategory β€” for instance, we have f(g(1R))=f(1S)=1Tf(g(1_R)) = f(1_S) = 1_T.

Ring-structure : βˆ€ β„“ β†’ Thin-structure β„“ Ring-on
Ring-structure β„“ .is-hom f x y = el! (is-ring-hom x y f)
Ring-structure β„“ .id-is-hom .pres-id = refl
Ring-structure β„“ .id-is-hom .pres-+ x y = refl
Ring-structure β„“ .id-is-hom .pres-* x y = refl
Ring-structure β„“ .∘-is-hom f g Ξ± Ξ² .pres-id = ap f (Ξ² .pres-id) βˆ™ Ξ± .pres-id
Ring-structure β„“ .∘-is-hom f g Ξ± Ξ² .pres-+ x y = ap f (Ξ² .pres-+ x y) βˆ™ Ξ± .pres-+ _ _
Ring-structure β„“ .∘-is-hom f g Ξ± Ξ² .pres-* x y = ap f (Ξ² .pres-* x y) βˆ™ Ξ± .pres-* _ _
Ring-structure β„“ .id-hom-unique Ξ± Ξ² i .Ring-on.1r = Ξ± .pres-id i
Ring-structure β„“ .id-hom-unique Ξ± Ξ² i .Ring-on._*_ x y = Ξ± .pres-* x y i
Ring-structure β„“ .id-hom-unique Ξ± Ξ² i .Ring-on._+_ x y = Ξ± .pres-+ x y i
Ring-structure β„“ .id-hom-unique {s = s} {t} Ξ± Ξ² i .Ring-on.has-is-ring =
  is-prop→pathp
    (Ξ» i β†’ hlevel {T = is-ring (Ξ± .pres-id i)
      (Ξ» x y β†’ Ξ± .pres-* x y i) (Ξ» x y β†’ Ξ± .pres-+ x y i)} 1)
    (s .Ring-on.has-is-ring) (t .Ring-on.has-is-ring) i

Rings : βˆ€ β„“ β†’ Precategory (lsuc β„“) β„“
Rings _ = Structured-objects (Ring-structure _)
module Rings {β„“} = Cat.Reasoning (Rings β„“)

Ring : βˆ€ β„“ β†’ Type (lsuc β„“)
Ring β„“ = Rings.Ob

In componentsπŸ”—

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 βˆ—* and ++, such that βˆ—* distributes over ++ on either side; ++ is an abelian group; and βˆ—* is a monoid.

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

    -- R is an abelian group:
    0R      : R
    _+_     : R β†’ R β†’ R
    -_      : R β†’ R
    +-idl   : βˆ€ {x} β†’ 0R + x ≑ x
    +-invr  : βˆ€ {x} β†’ x + (- x) ≑ 0R
    +-assoc : βˆ€ {x y z} β†’ (x + y) + z ≑ x + (y + z)
    +-comm  : βˆ€ {x y} β†’ x + y ≑ y + x

    -- R is a commutative monoid:
    1R      : R
    _*_     : R β†’ R β†’ R
    *-idl   : βˆ€ {x} β†’ 1R * x ≑ x
    *-idr   : βˆ€ {x} β†’ x * 1R ≑ x
    *-assoc : βˆ€ {x y z} β†’ (x * y) * z ≑ x * (y * z)

    -- Multiplication is bilinear:
    *-distribl : βˆ€ {x y z} β†’ x * (y + z) ≑ (x * y) + (x * z)
    *-distribr : βˆ€ {x y z} β†’ (y + z) * x ≑ (y * x) + (z * x)

This data is missing (by design, actually!) one condition which we would expect: 0β‰ 10 \ne 1. We exploit this to give our first example of a ring, the zero ring, which has carrier set the unit β€” 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 Z\mathbb{Z}, which is very far (infinitely far!) from having a single element. It’s called the β€œzero ring” because it has one element xx, which must be the additive identity, hence we call it 00. But it’s also the multiplicative identity, so we might also call the ring {βˆ—}\{*\} the One Ring, which would be objectively cooler.

Zero-ring : Ring lzero
Zero-ring = to-ring {R = ⊀} λ where
  .make-ring.ring-is-set _ _ _ _ _ _ β†’ tt
  .make-ring.0R β†’ tt
  .make-ring._+_ _ _ β†’ tt
  .make-ring.-_  _ β†’ tt
  .make-ring.+-idl _ β†’ tt
  .make-ring.+-invr _ β†’ tt
  .make-ring.+-assoc _ β†’ tt
  .make-ring.+-comm _ β†’ tt
  .make-ring.1R β†’ tt
  .make-ring._*_ _ _ β†’ tt
  .make-ring.*-idl _ β†’ tt
  .make-ring.*-idr _ β†’ tt
  .make-ring.*-assoc _ β†’ tt
  .make-ring.*-distribl _ β†’ tt
  .make-ring.*-distribr _ β†’ 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 Z\mathbb{Z}, 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:0β†’Rh : 0 \to R unless 0=h(0)=h(1)=10 = h(0) = h(1) = 1 in RR.

β„€ : Ring lzero
β„€ = to-ring {R = Int} Ξ» where
  .make-ring.ring-is-set β†’ hlevel 2
  .make-ring.0R β†’ 0
  .make-ring._+_ β†’ _+β„€_
  .make-ring.-_ β†’ negate
  .make-ring.+-idl β†’ +β„€-zerol _
  .make-ring.+-invr {x} β†’ +β„€-inverser x
  .make-ring.+-assoc {x} {y} {z} β†’ +β„€-associative x y z
  .make-ring.+-comm {x} {y} β†’ +β„€-commutative x y
  .make-ring.1R β†’ 1
  .make-ring._*_   β†’ _*β„€_
  .make-ring.*-idl β†’ *β„€-idl _
  .make-ring.*-idr β†’ *β„€-idr _
  .make-ring.*-assoc {x} {y} {z} β†’ *β„€-associative x y z
  .make-ring.*-distribl {x} {y} {z} β†’ *β„€-distrib-+β„€-l x y z
  .make-ring.*-distribr {x} {y} {z} β†’ *β„€-distrib-+β„€-r x y z

  1. being homomorphisms for the additive group, they automatically preserve zeroβ†©οΈŽ