Prove that a finitely generated module over a PID is projective if and only if it is free.
We begin with a lemma. (I probably already proved this somewhere, but I can’t find it now.)
Lemma: Let and be modules over a domain . Then . Proof: If , then we have for some nonzero . So and , and we have . If , then for some nonzero , we have . Since is a domain, is nonzero, and now . So .
Let be a finitely generated module over a principal ideal domain .
If is projective, then is free over for some module , by Proposition 30 on page 389 of D&F. (This is one of the equivalent definitions of projectivity for modules.) If is free, then it is torsion free, and we have by the lemma. So . By Theorem 5 in D&F (FTFGMPID), we have that for some , so that is free. By Corollary 31 on page 390 of D&F, (every free module is projective), is projective.
So for finitely generated modules over a PID, free and projective are equivalent.