Exercise 6, § 5

Let $A$ be an integral domain, $K$ its field of fractions and $M$ an $A$-module. Show that the following properties are equivalent: (a) $M$ is a projective $A$-module such that $[M \otimes_A K : K]$ is finite. (b) $M$ is finitely generated projective $A$-module. (y) $M$ is finitely generated and, for every maximal ideal $m$ of $A$, the $A_m$-module $M$ is free. (To show that (a) implies (b), use Algebra, Chapter 11, § 5, no. 5, Proposition 9. To show that (y) implies (a), observe first that $M$ is torsion-free and conclude that, for every prime ideal $p$ of $A$, $M$ is a free $A_p$-module of constant rank.)