The page 329 of the book "Modular Forms and Fermat Last Theorem" says that:
I have two questions about this equivalence.
In the cited reference, the definition further requests that the set of associated prime idals of $I$ is $\{\mathfrak{M}\}$ (if I understand correctly), but I don't think this can be concluded from (ii).
Also in the cited reference, it says that the functor ${\rm Hom}_R(-,I)$ actually gives an anti-isomorphism between the category of finitely generated $\hat{R}$-modules and the category of Artinian $R$-modules, but in (i) $\hat{R}$ is replaced by $R$. Since $I$ is Artinian as $R$-module and ${\rm Hom}_R(I,I)=\hat{R}$, this replacement holds true only when $\hat{R}$ is a finitely generated $R$-module. This seems to be true only in rare cases.
