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Let $R$ be a Dedekind domain, let $I$ be fractional ideal of $R$. Let $M$ be torsion free $R$module. Then I want to prove $I^{-1}M \cong Hom_R(I,M)$.

Let $ \phi : I^{-1}M→Hom_R(I,M)$ be given by $x→\phi_x:α→αx$. This is clearly homomorphism and injection because if $αx=αy$,$α(x-y)=0$ thus $x=y$ because $M$ is torsion free. I'm having trouble to prove surjectivity.

If this is an famous proposition, I'll also appreciated if you could tell me reference. Thank you for your help.

Pont
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    One approach is to prove this is an isomorphism $\mathfrak p$-locally for any prime ideal $\mathfrak p$ of $M$. That is, prove that the homomorphism $I^{-1}M\otimes_RR_{\mathfrak p}\to Hom_R(I,M)\otimes_RR_{\mathfrak p}$ is an isomorphism for each prime $\mathfrak p$. – Kenta S Sep 25 '22 at 18:45
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    This question is also answered here: https://math.stackexchange.com/questions/1344749/operatornamehom-r-mathfraka-m-is-isomorphic-to-mathfraka-1m-if?rq=1 – Kenta S Sep 25 '22 at 18:47
  • No, it's link gives just reference but reference is not valid now. – Pont Sep 26 '22 at 03:16
  • Reference link is invalid now. – Pont Sep 26 '22 at 03:28

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