Page "Preadditive category" Paragraph 10

Focusing on a single object A in a preadditive category, these facts say that the endomorphism hom-set Hom ( A, A ) is a ring, if we define multiplication in the ring to be composition.

This ring is the endomorphism ring of A. Conversely, every ring ( with identity ) is the endomorphism ring of some object in some preadditive category.

Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom ( A, A ) be R, and let composition be ring multiplication.

Since R is an Abelian group and multiplication in a ring is bilinear ( distributive ), this makes R a preadditive category.

Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object ( in the same way that a monoid can be viewed as a category with only one object-and forgetting the additive structure of the ring gives us a monoid ).