Pre-additive categories

Previously, we talked about groups with additonal structure. A more general concept is that of a category with additional structure, the most important example of which is a pre-additive category. Pre-additive categories are categories whose hom classes are commutative groups, such that the group distributes over composition. A (not necessarily commutative) ring is a pre-additive category with a single object.

Non-commutative rings of actions:
Let $C$ be a concrete category and $X \in Ob(C)$ then $End(X)$ is a monoid action on the underlying set of $X$. A wide variety of different monoid actions emerge from concrete categories like posets, graphs, etc. We can also explore the endomorphism monoid of a commutative group. In this case, pre-additive categories add additional perspective to the theory of commutative groups.

By taking the category of commutative groups to be an abelian category, we get that $End(G)$ for any commutative group is an endomorphism ring. Therefore the actions which previously only took on the structure of a monoid, now have the structure of a ring. This is the basic context in which non-commutative rings emerge and become indispensible. We naturally consider rings to be commutative starting with the ring of integers $\mathbb{Z}$, but here we see how some non-commutative rings emerge even in problems of commutative algebra.

Let $R$ be a ring, then the category of $R$-modules is an abelian category. In this case, for any $R$-module $M$ we have that $End(M)$ is the matrix ring of the module $M$. A more familiar case might be the matrix ring of a vector space over a field. These matrix rings are amongst the most important non-commutative rings, and they emerge as rings of actions on a set. Therefore, non-commutative rings are an indispensible part of the modern algebraic theory of actions.

Concrete rings:
A concrete pre-additive category is a pre-additive category $C$ with a faithful set-valued functor $F : C \to Sets$. This makes it so that all the elements of the ring are functions acting on a set. In this case, we have a multiplicative monoid action on the underlying set of the concrete ring. We see that the matrix ring $End(V)$ is a concrete ring whose elements are functions acting on the underlying set of vectors. Concrete rings proide a natural categorical description of non-commutative rings of actions.

Links:
Preadditive and additive categories