I am trying to figure out the following: Given a principal ideal domain $R$ which is not a field, does there necessarily exist a module over $A$ which is not free?
$A$ is a PID, so taking submodules won't work. The only approach I could come up with is taking some maximal ideal $m$ and considering the quotient field $A/m$. If $A/m$ were free, then its rank would be at most one, since we have an epimorphism $A\to A/m$. Hence the problem amounts to showing that $A$ and $A/m$ are not isomorphic as $A$-modules. Is this true? If so, does anyone know how to prove this? If not, can someone please provide a counterexample?
Thanks a lot,