Projective modules are direct summands of free modules

A module is projective iff it is a direct summand of a free module.

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

F \cong P \oplus Q

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).