Corollary

Let K be a commutative field, let $A_1$ and $A_2$ be K-algebras, and let $M_1$ be an absolutely semisimple $A_1$-module and $M_2$ a semisimple $A_2$-module. Then $M_1\otimes_KM_2$ is a semisimple module over the ring $A_1\otimes_KA_2$.

The module $M_1$ is the direct sum of absolutely semisimple simple $A_1$-modules (Proposition 4). It therefore suffices to prove the assertion in the case when the modules $M_1$ and $M_2$ are simple. Denote their commutants by $D_1$ and $D_2$. The K-algebra $D_1$ is absolutely semisimple (loc. cit.); by Corollary 1 of VIII, p. 234, the K-algebra $D_1\otimes_KD_2$ is semisimple. It then follows from Corollary 1 of VIII, p. 215 that the $(A_1\otimes_KA_2$)-module $M_1\otimes_KM_2$ is semisimple.