projective module

A module $P$ is projectiveif it satisfies the following equivalent conditions:

(a) Every short exact sequenceof the form $0\to A\to B\to P\to 0$is split (http://planetmath.org/SplitShortExactSequence);

(b) The functor $\mathrm{Hom}(P,-)$is exact (http://planetmath.org/ExactFunctor);

(c) If $f:X\to Y$ is an epimorphismand there exists a homomorphism $g:P\to Y$,then there exists a homomorphism $h:P\to X$such that $fh=g$.

(d) The module $P$ is a direct summand of a free module.