# injective module

A module $Q$ is an 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 split (http://planetmath.org/SplitShortExactSequence);

(b) The functor ${\rm Hom}(-,Q)$ is exact (http://planetmath.org/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 $hf=g$.

 $\xymatrix{0\ar[r]&X\ar[d]_{g}\ar[r]^{f}&Y\ar@{-->}[dl]^{h}\\ &Q}$
Title injective module InjectiveModule 2013-03-22 12:02:26 2013-03-22 12:02:26 antizeus (11) antizeus (11) 8 antizeus (11) Definition msc 16D50