Let $R$ be a principal ideal domain and $M$ a finitely generated $R$ module. Prove that $M$ is a free $R$-module if and only if $M$ is a projective $R$-module.
I am quite confused and totally not clear about projective modules defined by the universal lift property in commutative diagram.
Best Answer
In general, over any ring, for any module, the following implications hold:
free $\implies$ projective $\implies$ flat $\implies$ torsionfree
For finitely generated modules over a PID, the structure theorem says such a module is a direct sum of a free module and a torsion module. Thus, for such modules, torsionfree implies free, so all 4 properties are equivalent.