Projective module

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 $f\circ h=g$.

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 $n>1$.