Remark 1

Tag 00E9, permanent. Cite it as [Bourbaki, Tag 00E9].

Algebra, VIII, p. 213, § 12. Tensor Products of Semisimple Modules

Assertion a) of Theorem 1 is no longer true when the field K is not assumed algebraically closed. We can give examples (VIII, p. 225, Exercise 4) where $M_i$ is a simple $A_i$ -module that is finite-dimensional over K, for $i\in \{1,2\}$ , and where the A-module $M_1\otimes M_2$ is not semisimple or is semisimple but not simple.

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

Nothing.

Cites

VIII, p. 225, Exercise 4
Theorem 1