Splitting lemma

Splitting lemma: Given a short exact sequence

of $R$-modules, the following are equivalent:

1. There exists a module homomorphism $s\colon C\to B$ with $p\circ s=\mathrm{id}_C$ (a section of $p$).
2. There exists a homomorphism $r\colon B\to A$ with $r\circ i=\mathrm{id}_A$ (a retraction of $i$).
3. The sequence is split exact, i.e. $B\cong A\oplus C$ as a direct sum.

This lemma is the basic bridge between homomorphisms that admit one-sided inverses and internal direct sum decompositions.