Corollary

Tag 061T, permanent. Cite it as [Bourbaki, Tag 061T].

Algebra, chapter III, § 5. Tensor algebra. Tensors

If $M$ is a projective $A$ -module, $T(M)$ is a projective $A$ -module.

$M$ is a direct factor of a free $A$ -module $N$ (II, § 2, no. 2, Proposition 4) and hence $T(M)$ is a direct factor of $T(N)$ (no. 2); as $T(N)$ is free (Theorem 1), this shows that $T(M)$ is projective (II, § 2, no. 2).

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

Cited by

Corollary, § 6
Corollary 2, § 7

Cites

II, § 2, no. 2
II, § 2, no. 2, Proposition 4
Theorem 1