I have been trying to work out a proof of structure theorem for finitely generated modules over PID's using localization. This is what my plan is:

1.Prove that every finitely generated torsion free module over a PID is free.

2.Consider the canonical map $f:M\rightarrow M/M_{tor}$. It is easy to see that $M/M_{tor}$ is torsion free, so using (1) it is free(call it $F$). It turns out that $M$ is isomorphic to $kerf\bigoplus F$ and since $kerf$ is $M_{tor}$, $M$ is $M_{tor} \bigoplus F$.

So if (1) is done the target is almost fulfilled except for the structure of $M_{tor}$.

I am able to prove that if $E$ is a finitely generated torsion free module over a PID(say $R$) and $S^{-1}E$ is isomorphic to $K$(the quotient field of $R$) then $E$ is isomorphic to $R$. Where $S = R-{0}$

Now, if we can find a sub-module $L$ of $E$ such that $S^{-1}(E/L)$ is isomorphic to $K$ then I can proceed by induction to prove (1).

So any ideas how to find this $L$?

Thanks

minimalset of generators of $E$. That way $E/E^\prime$ can be generated with one less than number of generators of $E$. – Mahdi Majidi-Zolbanin Feb 28 '12 at 17:02