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).
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).