Proposition 6

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

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

Let $A_1$ and $A_2$ be nonzero K-algebras. If the ring $A_1\otimes A_2$ is simple (resp. semisimple), then the rings $A_1$ and $A_2$ are simple (resp. semisimple).

A ring B is semisimple (resp. simple) if and only if the B-module $B_s$ is semisimple (resp. isotypical and nonzero). The proposition therefore follows from Proposition 1 (VIII, p. 211).

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

Remark 1, § 14

Cites

Proposition 1 (VIII, p. 211