Is the following short exact sequence always split?

Question

Is it always true that the following short exact sequence of modules split?

$0\to N\to M\to M/N\to0$

Answer 1

This need not split. As an example, take $0 \to 2\mathbb{Z} \hookrightarrow \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$. More generally, split short exact sequences are precisely those that admit a decomposition $M \approx N \oplus M/N$, which need not be true in general.