Let $R = \mathbb{Z}[i]/(5)$.
It is obvious that $R$ is not an integral domain, and any ideal in $R$ is principal.
Now I want to prove the following classification theorem for modules over $R$ :
There exist modules $M_1, M_2$ such that any finitely generated module $M$ over $R$ is isomorphic to the direct sum $ M_1^r ⊕ M_2^s$, where $M_1^r$ is the direct sum of r copies of module $M_1$, and similarly for $M_2$ .