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Math Help - Modules over P.I.D

  1. #1
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    Modules over P.I.D

    Show that any finitely generated torsion free module is a free module..

    Any help is appreciated ..
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  2. #2
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    Quote Originally Posted by math.dj View Post
    Show that any finitely generated torsion free module is a free module..

    Any help is appreciated ..
    See Hungerford's algebra book p221.

    The sketch of the proof is

    Let A be a finitely generated torsion-free module over PID R. Let X be a finite set of nonzero generators of A. Let S be a maximal subset of X such that the submodule F generated by the inclusion i:S \rightarrow X is free. For each y_i \in X \setminus S, there exists 0 \neq r_i \in R such that r_iy_i \in F. Let r=\prod_i r_i. It follows that rA \subset F. Since A is torsion-free, f:A \rightarrow A given by a \mapsto ra is a R-module homomorphism whose kernel is trivial. So, A \cong rA. Verify that rA is a submodule of A and a submodule of a free R-module for PID R is free. Thus, A is free.
    Last edited by TheArtofSymmetry; January 18th 2011 at 04:43 AM.
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