proof of finitely generated torsion-free modules over Prüfer domains

Let M be a finitely generatedMathworldPlanetmathPlanetmathPlanetmath torsion-free module over a Prüfer domain R with field of fractionsMathworldPlanetmath k. We show that M is isomorphicPlanetmathPlanetmathPlanetmath to a direct sumMathworldPlanetmathPlanetmathPlanetmath ( of finitely generated ideals in R.

We shall write kM for the vector spaceMathworldPlanetmath over k generated by M. This is just the localizationMathworldPlanetmath ( of M at R{0} and, as M is torsion-free, the natural map MkM is one-to-one and we can regard M as a subset of kM.

As M is finitely generated, the vector space kM will finite dimensional (, and we use inductionMathworldPlanetmath on its dimensionPlanetmathPlanetmath n. Supposing that n>0, choose any basis e1,,en and define the linear map f:kMk by projectionPlanetmathPlanetmathPlanetmath ( onto the first componentMathworldPlanetmathPlanetmath,


Restricting to M, this gives a nonzero map Mk. Furthermore, as M is finitely generated, f(M) will be a finitely generated fractional idealPlanetmathPlanetmath in k. Choosing any nonzero cR such that 𝔞cf(M)R,


defines a homorphism from M onto the nonzero and finitely generated ideal 𝔞. As R is Prüfer and invertible ideals are projective, g has a right-inverse h:𝔞M. Then h has the left-inverse g and is one-to-one, so defines an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between 𝔞 and its image ( We decompose M as the direct sum of the kernel of g and the image of h,


Projection from the finitely generated module M onto ker(g) shows that it is finitely generated and,


So, the result follows from applying the induction hypothesis to ker(g).

Title proof of finitely generated torsion-free modules over Prüfer domains
Canonical name ProofOfFinitelyGeneratedTorsionfreeModulesOverPruferDomains
Date of creation 2013-03-22 18:36:14
Last modified on 2013-03-22 18:36:14
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 13F05
Classification msc 13C10