Proposition 4

Let P be an A-module. The following properties are equivalent:

(a) P is projective. (b) For every exact sequence 0 → F' → F → F'' → 0 of A-linear mappings, the sequence

is exact.

(c) For every surjective A-module homomorphism u : E → E'' and every homomorphism f : P → E'', there exists a homomorphism g : P → E such that f = u ∘ g (it is said that f can be "lifted" to a homomorphism of P into E).

(d) Every exact sequence 0 → E' → E → P → 0 of A-linear mappings splits (and therefore P is isomorphic to a direct factor of E).

(e) P is isomorphic to a direct factor of a free A-module.

It is trivial that (a) implies (b). To see that (b) implies (c), it suffices to apply (b) to the exact sequence 0 → E' → E → E'' → 0, where E' = Ker(u), since (c) expresses the fact that

is surjective. To see that (c) implies (d), it suffices to apply (c) to the surjective homomorphism $v : E \to P$ and the homomorphism $l_P : P \to P$; the existence of a homomorphism $g : P \to E$ such that $l_P = v \circ g$ implies that the sequence

splits (§ 1, no. 9, Proposition 15). As for every A-module M there exist a free A-module L and an exact sequence $0 \to R \to L \to M \to 0$ (§ 1, no. 11, Proposition 20), clearly (d) implies (e). Finally (e) implies (a) by virtue of Proposition 3 and its Corollary.