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A module $Q$ is an 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 split;
(b) The functor ${\rm Hom}(-, Q)$ is exact;
(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 $hf = g$ .
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![$\displaystyle \xymatrix{ 0 \ar[r] & X \ar[d]_g \ar[r]^f & Y \ar@{-->}[dl]^h \ & Q } $](http://images.planetmath.org:8080/cache/objects/1083/js/img1.png "$\displaystyle \xymatrix{ 0 \ar[r] & X \ar[d]_g \ar[r]^f & Y \ar@{-->}[dl]^h \ & Q } $")