# 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)

$$M\cong {\U0001d51e}_{1}\oplus \mathrm{\cdots}\oplus {\U0001d51e}_{n}$$ |

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).

Title | finitely generated torsion-free modules over Prüfer domains |
---|---|

Canonical name | FinitelyGeneratedTorsionfreeModulesOverPruferDomains |

Date of creation | 2013-03-22 18:36:11 |

Last modified on | 2013-03-22 18:36:11 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 13F05 |

Classification | msc 13C10 |

Related topic | EquivalentCharacterizationsOfDedekindDomains |