Finitely generated torsion-free over a PID is free

Finitely generated torsion-free over a PID is free: Let $R$ be a PID and let $M$ be a finitely generated torsion-free $R$-module. Then $M$ is a free $R$-module.

This is the torsion-free specialization of the structure theorem for finitely generated modules over a PID : for a PID , a finitely generated torsion-free module has no torsion summand and must therefore be free .