Examples: Representations over finite rings and Maschke's theorem

Question

Is there a possibility to get the simple $R[G]$-modules, if $R$ is the ring $\mathbb{Z}/n\mathbb{Z}$, $G$ a finite group and $\operatorname{ord}(G)$ and $n$ are relatively prime? For which groups would their number be finite?

I could solve this just for special $n$, but not in general. Maybe there is some literature where one can find examples for special groups?

Thanks and best regards.

Edit: Can one solve this maybe for $n$ a power of a prime?

Answer 1

Every ring with identity has simple modules: you just take $R/M$ where $M$ is a maximal right ideal (or left ideal, if you want a left module.)

If $G$ is finite then this group ring is Artinian, but this paper leads me to believe that not all Artinian rings are of finite representation type.

If $G$ is finite and the order $|G|$ is a unit in $R$, the ring is semisimple by Maschke's theorem, and hence has finite representation type.