# projective module

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

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

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

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

 $\xymatrix{&P\ar@{-->}[dl]_{h}\ar[d]^{g}\\ X\ar[r]_{f}&Y\ar[r]&0}$

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

Title projective module ProjectiveModule 2013-03-22 12:09:42 2013-03-22 12:09:42 antizeus (11) antizeus (11) 7 antizeus (11) Definition msc 16D40 InvertibleIdealsAreProjective