Corollary 1

Tag 00EO, permanent. Cite it as [Bourbaki, Tag 00EO].

Algebra, VIII, p. 222, § 12. Tensor Products of Semisimple Modules

a) Suppose that the A-module M is semisimple, that the extension L of K is separable, and that M or L is finite-dimensional over K. Then the $A_{(L)}$ -module $M_{(L)}$ is semisimple.

b) Suppose that the A-module M is simple and that its commutant is equal to K. Then the $A_{(L)}$ -module $M_{(L)}$ is simple.

Assertion a) follows from Corollary VIII, p. 221. Assertion b) is a specific case of Proposition 8, c).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Definition 1, § 13
Proposition 2, § 13

Cites

VIII, p. 221
Proposition 8