I am reading Lang's Algebra and trying to fill in the gaps in my mathematical background while I train for the quals. So I came across the following exercise (chapter 20, ex. 26 in the third edition).
$A$ is a commutative ring, $E$ an $A$-module and its dual is $E^*=\mathrm{Hom}_A(E; \mathbb Q/\mathbb Z)$.
It is asked to prove that a short sequence \begin{equation} 0 \rightarrow N \rightarrow M \rightarrow E \rightarrow 0 \end{equation} is exact iff the dual sequence \begin{equation} 0 \rightarrow E^* \rightarrow M^* \rightarrow N^* \rightarrow 0 \end{equation} is exact.
It is not hard to prove that the dual sequence is exact if the first one is, using the arguments Lang shows, in particular the injectivity of the module $\mathrm{Hom}_A(A; \mathbb Q/\mathbb Z)$. The vice versa turns out to be harder (at least, I can't find a clear way for that), besides the easy proof that $ N \rightarrow M$ is injective, thanks to the canonical embedding of a module $M$ in its bidual $M^{**}$. Any hint or reference? Also any interesting reading on the topics of injective and projective module, resolutions, etc, will be appreciated.
Thanks a lot.