A module is an injective module if it satisfies the following equivalent conditions:
(a) Every short exact sequence of the form is split;
(b) The functor is exact;
(c) If is a monomorphism and there exists a homomorphism , then there exists a homomorphism such that .
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ 0 \ar[r] & X \ar[d]_g \ar[r]^f & Y \ar@{-->}[dl]^h \ & Q } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/1083/l2h/img8.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ 0 \ar[r] & X \ar[d]_g \ar[r]^f & Y \ar@{-->}[dl]^h \ & Q } } \end{xy}$")