Projective short exact sequence criterion

Projective s.e.s. criterion: An $R$-module $P$ is projective if and only if every short exact sequence

is split exact.

This reformulates projectivity as a splitting property for extensions of $P$. The "only if" direction uses a lift of $\mathrm{id}_P$ along the surjection $B\to P$, and the "if" direction follows by encoding a lifting problem as the splitting of an induced extension (compare the splitting lemma).

Writing $p\colon B\to P$ for the surjection, lifting $\mathrm{id}_P$ along $p$ gives the section $s$, with $p\circ s=\mathrm{id}_P$: