An injective-injective module problem

Question

I want to prove this problem:

For an $R$-module $M$ and an ideal $J⊆R$, let $A=\{m∈M∶mJ=0\}$. If $M$ is an injective $R$-module, show that $A$ is an injective $R/J$-module.

We can view $A$ as an $R/J$-module, and letting $E=E(A_{R/J})$ (the injective hull of $A_{R/J}$) we write $A⊆_eE$ (meaning $A$ is essential in $E$) as $R/J$-module, and hence as $R$-modules. Now, it suffices to prove that $A=E$. Thanks for any cooperation.

Answer 1

Notice that $A \cong \hom_R(R/J,M)$. Now we have the following fact:

If $R \to S$ is a ring homomorphism and $M$ is an injective $R$-module, then $\hom_R(S,M)$ is an injective $S$-module.

In fact, $\hom_S(-,\hom_R(S,M)) \cong \hom_R((-)|_R,M)$ is exact.