proof of finitely generated torsion-free modules over Prüfer domains
Let be a finitely generated torsion-free module over a Prüfer domain with field of fractions . We show that is isomorphic to a direct sum (http://planetmath.org/DirectSum) of finitely generated ideals in .
We shall write for the vector space over generated by . This is just the localization (http://planetmath.org/LocalizationOfAModule) of at and, as is torsion-free, the natural map is one-to-one and we can regard as a subset of .
As is finitely generated, the vector space will finite dimensional (http://planetmath.org/Dimension2), and we use induction on its dimension . Supposing that , choose any basis and define the linear map by projection (http://planetmath.org/Projection) onto the first component,
Restricting to , this gives a nonzero map . Furthermore, as is finitely generated, will be a finitely generated fractional ideal in . Choosing any nonzero such that ,
defines a homorphism from onto the nonzero and finitely generated ideal . As is Prüfer and invertible ideals are projective, has a right-inverse . Then has the left-inverse and is one-to-one, so defines an isomorphism between and its image (http://planetmath.org/ImageOfALinearTransformation). We decompose as the direct sum of the kernel of and the image of ,
Projection from the finitely generated module onto shows that it is finitely generated and,
So, the result follows from applying the induction hypothesis to .
|Title||proof of finitely generated torsion-free modules over Prüfer domains|
|Date of creation||2013-03-22 18:36:14|
|Last modified on||2013-03-22 18:36:14|
|Last modified by||gel (22282)|