Let $A$ be a P.I.D. Let $K$ be it's field of fractions.
Let $N$ be any $A$-module.
Denote: $N^v := \mathrm{Hom}_A(N, K/A)$.
Let $M$ be a finitely generated torsion $A$-module.
Fix any $m\in M$.
Define $\psi_m : M^{v} \rightarrow K/A$ by $\psi_m(\phi) = \phi(m)$, where $\phi\in M^v$.
Define $\Psi : M \rightarrow (M^v)^v$ by $\Psi(m) = \psi_m$.
Show that $\Psi$ is an isomorphism.
I have already shown that $\psi$, $\Psi$ are both homomorphism, thus I believe that to prove the claim we just need to show that $\Psi$ is bijective.
But I'm stuck trying to prove it's injectivity and surjectivity.
Any help or insight is deeply appreciated.