Corollary 1

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

Algebra, VIII, p. 221, § 12. Tensor Products of Semisimple Modules

Let $A_1$ and $A_2$ be semisimple K-algebras; suppose that $A_1$ is finite-dimensional over K. Suppose that the center of $A_1$ or $A_2$ is a separable algebra over K, which is, for example, the case when K is perfect. Then $A_1\otimes A_2$ is semisimple.

This is the case $M_1= (A_1)_s, M_2= (A_2)_s$ of the corollary of VIII, p. 221.

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

Theorem 5, § 14

Cites

VIII, p. 221