Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ideal of $R$ and $M$ is an $R$-module, then is it true that $\text{Hom}_R(R/I, E^0_R(M))\cong E^0_{R/I}(\text{Hom}_R(R/I, M))$ ?
(2) If $M$ is an $R$-module and $S$ is a commutative Noetherian $R$-algebra such that $\text{Ext}_R^{>0}(S,M)=0$, then is it true that $E^i_S(\text{Hom}_R(S,M))\cong \text{Hom}_R(S, E^i_R(M))$ for all $i\ge 0$ ?
