projective module ProjectiveModule 2002-01-05 15:46:12 2003-09-20 21:38:32 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A module $P$ is {\it 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 \PMlinkname{split}{SplitShortExactSequence}; (b) The functor ${\rm Hom}(P, -)$ is \PMlinkname{exact}{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.