I was stuck with the seemingly simple homework problem:
A $R$-module is injective if and only if every exact sequence $$0\rightarrow E\rightarrow B\rightarrow R/I\rightarrow 0$$ splits.Here $I$ is an ideal of $R$.
The only if direction is straightforward since if $E$ is injective, then every exact sequence of former type must splits. But I do not know how to prove $E$ is injective given the above condition. The natural direction is to use Baer's criterion. However the above exact sequence seems to be quite difficult to apply this criterion. So I am stuck.