Proposition 3

Tag 05N3, permanent. Cite it as [Bourbaki, Tag 05N3].

Algebra, chapter II, § 2. Modules of linear mappings. Duality

For an A-module P, the direct sum of a family of submodules (M_i), to be projective, it is necessary and sufficient that each of the M_i be projective.

For every A-module homomorphism u : E → F,

\operatorname{Hom}(1_P, u) : \operatorname{Hom}(P, E) \to \operatorname{Hom}(P, F)

is identified with $\prod_i \operatorname{Hom}(1_{M_i}, u)$ (§ 1, no. 6, Corollary 1 to Proposition 6); the conclusion thus follows from Definition 1 and § 1, no. 5, Proposition 5 (ii).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 2
Proposition 4, § 2

Cites

§ 1, no. 5, Proposition 5
§ 1, no. 6, Corollary 1 to Proposition 6
Definition 1