See Hungerford's algebra book p221.

The sketch of the proof is

Let A be a finitely generated torsion-free module over PID R. Let X be a finite set of nonzero generators of A. Let S be a maximal subset of X such that the submodule F generated by the inclusion is free. For each , there exists such that . Let . It follows that . Since A is torsion-free, given by is a R-module homomorphism whose kernel is trivial. So, . Verify that rA is a submodule of A and a submodule of a free R-module for PID R is free. Thus, A is free.