Why can $M$ be an abelian group?

Question

In Rotman's, "An introduction to homological algebra", there is a proof of that the categories of modules has enough injectives.

In the proof, he proves that there is an injective map from $M$ as an abelian group to an injective module, why this proves that $M$ as a module can be embebed in such injective module?

Answer 1

Every left $R$-module can be seen as an $R$-$\Bbb Z$-bimodule, and any Abelian group as a $\Bbb Z$-$\Bbb Z$ bimodule.

Now, if ${}_AM_B$ and ${}_AN_C$ are bimodules, we get a $B$-$C$-bimodule structure on $\hom_A(M,N)$ by defining $$b\cdot f\cdot c\ :=\ m\mapsto f(mb)\cdot c\,.$$

It follows that $\hom({}_{\Bbb Z}R,\,{}_{\Bbb Z}D)$ is a left $R$-module by the right $R$-action on $R$, and they verify that the injective map $M\to\hom_{\Bbb Z}(R,M)\to\hom_{\Bbb Z}(R,D)$ actually preserves the left $R$-action.

Now, if we know that $\hom_{\Bbb Z}(R,D)$ is an injective $R$-module whenever $D$ is a divisible Abelian group, then we're done.