Find an example of a short exact sequence of Z-modules that does not split.
Any hints?
$\displaystyle 0 \to \mathbb Z \to\mathbb Q \to \mathbb{Q/Z} \to 0$. It is clearly a short exact sequence of abelian group and $\displaystyle \text{Hom}_\mathbb{Z}(\mathbb{Q/Z}, \mathbb{Q})=0$, since $\displaystyle \mathbb{Q/Z}$ is torsion while $\displaystyle \mathbb Q$ is torsion-free.