finitely generated torsion-free modules over Prüfer domains

Theorem.

Let $M$ be a finitely generated torsion-free module over a Prüfer domain $R$. Then, $M$ is isomorphic to a direct sum (http://planetmath.org/DirectSum)

$$M\cong {𝔞}_1\oplus \mathrm{\cdots }\oplus {𝔞}_n$$

of finitely generated ideals ${\mathrm{a}}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{\mathrm{a}}_n$.

As invertible ideals are projective and direct sums of projective modules are themselves projective, this theorem shows that $M$ is also a projective module. Conversely, if every finitely generated torsion-free module over an integral domain $R$ is projective then, in particular, every finitely generated nonzero ideal of $R$ will be projective and hence invertible. So, we get the following characterization of Prüfer domains.

Corollary.

An integral domain $R$ is Prüfer if and only if every finitely generated torsion-free $R$-module is projective (http://planetmath.org/ProjectiveModule).