# Finitely generated torsion module over a PID.

Let $A$ be a PID, $K$ be the field of fractions of $A$, and $M$ be a finitely generated torsion $A$-module. Let $M'=\text{Hom}(M,K/A)$ and $M''=\text{Hom}(M',K/A)$. I want to show that the evaluation homomorphism $M \to M''$ (given by $x \mapsto ev_x$) is an isomorphism.

I used the structure theorem and the fact $\text{Hom}(A/(a),K/A) \simeq A/(a)$, which showed me that $M \simeq M' \simeq M''$. But I don't know why the evaluation homomorphism is indeed an isomorphism. How can I show it?

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How would you show that, for a finite dimensional vector space $\rm V$, the evaluation map $\rm V \to V^{**}$ is an isomorphism ?

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There was a basis, dual basis to start with, and also used that in finite dimensional vector space, injectivity implies surjectivity. –  Gobi Mar 16 '13 at 1:03