Projective module - Knowlpedia

An $R$ -module $P$ is projective if for every surjective homomorphism $f:M\to N$ and every homomorphism $g:P\to N$ (see module homomorphisms), there exists a homomorphism $h:P\to M$ such that $f\circ h=g$ .

As a diagram, $g$ lifts along the surjection $f$ and the triangle commutes:

Equivalent characterizations

Equivalently, $P$ is projective iff it is a direct summand of a free module; see projective is a summand of free. Projectivity can also be detected via splitting of short exact sequences ending in $P$ (a standard criterion is the projective short-exact-sequence criterion), linking it to short exact sequences.

Examples

Every free module is projective (take lifts coordinatewise).
Any direct summand of a free module (e.g. $R^n \cong P\oplus Q$ ) is projective.
(Nonexample) As a $\mathbb Z$ -module, $\mathbb Z/n\mathbb Z$ is not projective for $n>1$ .