I did some googling and found a few references. First, this article reduces a problem the authors are interested in to a well known problem in the representation theory of finite rings. At the end are some references to that problem and those should include the necessary background you need to get into this subject.
Another reference is this MO post, which includes as many freely available texts on representation theory of groups as could be found. Perhaps one of them discusses or mentions representation theory of rings as well.
In case none of those references pan out, here is perhaps a place to start your thinking. A representation of a group is $\rho: G \rightarrow GL(\mathbb{C})$ and is equivalent to a module over the group ring $\mathbb{C}[G]$. A representation of a ring would need to be $\rho: R\rightarrow GL(\mathbb{C})$ and should be equivalent to a module $M$ over a more general ring. We can form det $\circ \rho$: from the units of $R$ to $\mathbb{C}$, and this restricts the form $M$ can take.