Preadditive Category - Elementary Properties

Elementary Properties

Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.

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).

In this way, preadditive categories can be seen as a generalisation of rings. Many concepts from ring theory, such as ideals, Jacobson radicals, and factor rings can be generalized in a straightforward manner to this setting. When attempting to write down these generalizations, one should think of the morphisms in the preadditive category as the "elements" of the "generalized ring". We won't go into such depth in this article.