InjectiveModule 2001-12-12 00:00:50 2004-03-11 00:41:09 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A module $Q$ is an {\it injective module} if it satisfies the following equivalent conditions: (a) Every short exact sequence of the form $0 \to Q \to B \to C \to 0$ is \PMlinkname{split}{SplitShortExactSequence}; (b) The functor ${\rm Hom}(-, Q)$ is \PMlinkname{exact}{ExactFunctor}; (c) If $f : X \to Y$ is a monomorphism and there exists a homomorphism $g : X \to Q$, then there exists a homomorphism $h : Y \to Q$ such that $hf = g$. $$ \xymatrix{ 0 \ar[r] & X \ar[d]_g \ar[r]^f & Y \ar@{-->}[dl]^h \\ & Q } $$