Corollary 1
Tag 0481, permanent. Cite it as [Bourbaki, Tag 0481].
Commutative Algebra, chapter II, § 3. Local rings. Passage from the local to the global
Let A be a not necessarily commutative ring, m the Jacobson radical of A and M a left A-module. Suppose that A/m is a field, that the canonical homomorphism from m \otimes_A M to M is injective and that one of conditions (i), (ii) of Proposition 5 is satisfied. For a family (y_\lambda) of elements of M to be a basis of a direct factor of M, it is necessary and sufficient that the family (1 \otimes y_\lambda) be free in M/mM.
If this condition holds, it can be assumed that (y_\lambda) is a subfamily of a family (x_i) of elements of M such that (1 \otimes x_i) is a basis of M/mM (Algebra, Chapter II, § 7, no. 1, Theorem 2) and Proposition 5 then proves that (x_i) is a basis of M.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.