open import Algebra.Ring

open import Cat.Abelian.Base
open import Cat.Prelude

module Cat.Abelian.Endo {o ℓ} {C : Precategory o ℓ} (A : Ab-category C) where

private module A = Ab-category A

Endomorphism rings🔗

Fix an $\mathbf{Ab}$-category $\mathcal{A}$: It can be the category of abelian groups $\mathbf{Ab}$ itself, for example, or $R$-Mod for your favourite ring¹. Because composition in $\mathcal{A}$ distributes over addition, the collection of endomorphisms of any particular object $x : \mathcal{A}$ is not only a monoid, but a ring: the endomorphism ring of $x$.

Endo : A.Ob → Ring ℓ
Endo x = to-ring mr where
  open make-ring
  mr : make-ring (A.Hom x x)
  mr .ring-is-set = A.Hom-set x x
  mr .0R = A.0m
  mr .1R = A.id
  mr .make-ring._+_ = A._+_
  mr .make-ring.-_ = A.Hom.inverse
  mr .make-ring._*_ = A._∘_
  mr .+-idl = A.Hom.idl
  mr .+-invr = A.Hom.inverser
  mr .+-assoc = A.Hom.associative
  mr .+-comm = A.Hom.commutes
  mr .*-idl = A.idl _
  mr .*-idr = A.idr _
  mr .*-assoc = A.assoc _ _ _
  mr .*-distribl = sym (A.∘-linear-r _ _ _)
  mr .*-distribr = sym (A.∘-linear-l _ _ _)

This is a fantastic source of non-commutative rings, and indeed the justification for allowing them at all. Even for the relatively simple case of $\mathcal{A} = \mathbf{Ab}$, it is an open problem to characterise the abelian groups with commutative endomorphism rings.