Let $R$ be a one-dimensional domain and $I \subset R$ an integral ideal. Is its dual module $I^\vee = \operatorname{Hom}_R(I,R)$ isomorphic to a fractional ideal of $R$?
If $I$ is invertible, then we have $I^\vee \cong I^{-1}$ which is a fractional ideal. But what if $I$ is not invertible?
I would like to see an easy counter-example with a $R$ one-dimensional but not Dedekind! This comes down to see such a homomorphism which is not given by multiplication with an element of $\operatorname{Quot}(R)$.
Thank you in advance!