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≅𝔞1⊕⋯⊕𝔞n |
of finitely generated ideals a1,…,an.
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 |
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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 |