I am new to category theory. I understand that a monoid $(M,+)$ can be represented as a category $C$ with the class of objects being the singleton {$M$}, the morphisms of $C$ corresponding to elements of M, the composition of those morphisms corresponding to + on $M$, and the identity morphism of $M$ being the identity element in $(M,+)$.
Question: Can a ring $(R,+,*)$ be represented in a similar way by a single category or some combination of several categories?