Let $R$ be a principal ideal domain with $|R|=\infty$. Suppose $M$ is an $R$-module such that $2 \le |M| < \infty$. What properties does this imply about $R$?
Background: I was hoping that $R\cong \mathbb{Z}$ is the only possibility, since $M$ is (also) a finitely generated $\mathbb{Z}$-module.
If I understand well you have an infinite PID which has a finite non-zero module, say $M$. In particular, $M$ is finitely generated and torsion, so it is a (finite) direct sum of indecomposable torsion modules, that is, of modules of the form $R/(p^m)$ with $p\in R$ prime, $p\ne0$. This shows that you have to impose on $R$ the following condition:
which is equivalent to $R/(p^m)$ is finite for some prime $p\in R$, $p\ne0$ and $m\ge1$.
For instance, $R=\mathbb Z[i]$ is such an example (other than $\mathbb Z$).