Projective modules are direct summands of free modules

Projective is a summand of free: An $R$-module $P$ is projective if and only if there exists a free module $F$ and an $R$-module $Q$ such that

as a direct sum.

This characterization is often the most practical: projective modules are precisely the modules that can be “split off” from free modules (equivalently, retracts of free modules).

In the retract diagram below, $s$ and $\pi$ are the inclusion and projection of the summand $P$; conversely, any pair with $\pi\circ s=\mathrm{id}_P$ gives $F=s(P)\oplus\ker\pi$: