Let $R$ be a principal ideal domain. Prove that any submodule of a finitely-generated free module over $R$ is free.

An $R$-module $P$ is said to be projective if, whenever we have module homomorphisms $f: M \rightarrow N$ and $g: P \rightarrow N$ with $f$ surjective, there exists a homomorphism $h: P \rightarrow M$ with $f \circ h=g$. Show that any free module (over an arbitrary ring) is projective. Show also that a finitely-generated projective module over a principal ideal domain is free.