Submodule of a free module over a principal ring

Question

I'm looking a counterexample.

A free module over a principal ring such that it exists a submodule
which is not free. (Of course it's impossible when the module is
finitely generated).

Thanks and regards.

EDIT: The main problem I have is to prove that $Z^{N}$ or $Z_{2}^{N}$ are free. Otherwise I think they could be a good couterexemple. Maybe it's possible to find easier solution.

Answer 1

You can't find such a counterexample since it is a direct consequence of the following theorem:

Let $A$ be a ring such that every left ideal is projective. Every submodule $M$ of a free left $A$-module $L$ is a direct sum of modules isomorphic to ideals of $A$.

(Bourbaki, Algebra, ch.7, Modules over Principal Ideal Domains, §3 theorem 1).

Another consequence is that a submodule of a projective module over a Dedekind domain is projective.