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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.

EDIT: For any commutative ring with identity $R$, the $R$-module $E$ is said to be an injective cogenerator if the functor $Hom_{R}(-,E)$ is exact and faithful; this is, $E$ is an injective $R$-module such that, given any non-zero element $x$ in a module $M$, there exists a homomorphism of $R$-modules $\varphi:M\rightarrow E$ such that $\varphi(x)\neq0$. A minimal injective cogenerator of $R$ is an injective cogenerator which is contained in any injective cogenerator. If $R$ is Noetherian, the minimal injective cogenerator of $R$ can be expressed as the module $\bigoplus E_{R}(R/\mathfrak{m})$ where $\mathfrak{m}$ runs on all the maximal ideals of $R$.

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.

EDIT: For any commutative ring with identity $R$, the $R$-module $E$ is said to be an injective cogenerator if the functor $Hom_{R}(-,E)$ is exact and faithful; that is is, $E$ is an injective $R$-module such that, given any non-zero element $x$ in a module $M$, there exists a homomorphism of $R$-modules $\varphi:M\rightarrow E$ such that $\varphi(x)\neq0$. A minimal injective cogenerator of $R$ is an injective cogenerator which is contained in any injective cogenerator. If $R$ is Noetherian, the minimal injective cogenerator of $R$ can be expressed as the module $\bigoplus E_{R}(R/\mathfrak{m})$ where $\mathfrak{m}$ runs over all the maximal ideals of $R$.

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.

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.

EDIT: For any commutative ring with identity $R$, the $R$-module $E$ is said to be an injective cogenerator if the functor $Hom_{R}(-,E)$ is exact and faithful; this is, $E$ is an injective $R$-module such that, given any non-zero element $x$ in a module $M$, there exists a homomorphism of $R$-modules $\varphi:M\rightarrow E$ such that $\varphi(x)\neq0$. A minimal injective cogenerator of $R$ is an injective cogenerator which is contained in any injective cogenerator. If $R$ is Noetherian, the minimal injective cogenerator of $R$ can be expressed as the module $\bigoplus E_{R}(R/\mathfrak{m})$ where $\mathfrak{m}$ runs on all the maximal ideals of $R$.