Is torsion-free equivalent to free for non-finitely generated modules over a PID?

Question

Maybe this is a trivial question. If $A$ is a PID and $M$ is a finitely generated $A$-module, it's well known that $M$ is torsion-free iff $M$ is free.

However, if $M$ is not finitely generated, does the result still hold? I think that the answer is no, simply because I could't find it in any reference.

If the answer is negative, please, show a counter-example.

Thanks in advance.

Answer 1

A counterexample is $\mathbb{Q}$ considered as a $\mathbb{Z}$-module. Definitely torsion-free but not free.