A module $P$ is projective if it satisfies the following equivalent conditions:

(a) Every short exact sequence of the form $0 \to A \to B \to P \to 0$ is split;

(b) The functor ${\rm Hom}(P, -)$ is exact;

(c) If $f : X \to Y$ is an epimorphism and there exists a homomorphism $g : P \to Y$ , then there exists a homomorphism $h : P \to X$ such that $fh = g$ .

(d) The module $P$ is a direct summand of a free module.