Assume that $R$ is a commutative Noetherian ring with minimal injective cogenerator $E$. For a finite set of maximal ideals $X$ of $R$, define the multiplicative set $$S_X=R-\bigcup_{\mathfrak{m}\in X}\mathfrak{m}.$$

The question is: $$Hom_R(S_X^{-1}R, E) =~\!\! ?$$

I know that this module is an injective cogenerator of $S_X^{-1}R$. I want to know if it is its minimal one.