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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?

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Yes, just as a monoid is a one object category, a unitary ring can be a one object pre-additive category, i.e., a category enriched over the Abelian groups, that is there is an Abelian group structure on each homset, such that the composition preserves the Abelian group operations $+,-$.

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We assume that $R$ has an identity. Since $R$ is a multiplicative monoid, $R$ can be regarded as a category. It is a preadditive category.

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