# 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)

 $M\cong\mathfrak{a}_{1}\oplus\cdots\oplus\mathfrak{a}_{n}$

of finitely generated ideals $\mathfrak{a}_{1},\ldots,\mathfrak{a}_{n}$.

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).

Title finitely generated torsion-free modules over Prüfer domains FinitelyGeneratedTorsionfreeModulesOverPruferDomains 2013-03-22 18:36:11 2013-03-22 18:36:11 gel (22282) gel (22282) 4 gel (22282) Theorem msc 13F05 msc 13C10 EquivalentCharacterizationsOfDedekindDomains