injective module

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 $hf = g$ .