Projective modules are direct summands of free modules ($P\text{ projective}⟺P\text{ is a direct summand of a free module}$)

Every module $P$ is a quotient of a free module $F={⨁}_{p\in P}A{e}_p$, via $e_p\mapsto p$. If $P$ is a projective module, lift $1_P$ through this surjection to obtain a section $s:P\to F$. The resulting split short exact sequence identifies $F$ with $\ker \pi \oplus P$. Conversely, if $P$ is a summand of a free module, extend a prescribed map out of $P$ to the free module using the summand projection, lift each basis image through the given surjection, and restrict the lift to $P$. This proves the equivalence with the lifting definition of a projective module, for arbitrary ranks.