Proposition 7

Let $A_1$ and $A_2$ be semisimple K-algebras, with respective centers $Z_1$ and $Z_2$. Suppose that $A_1$ has finite degree over K. Then the ring $A_1\otimes A_2$ is left Artinian, as is its center $Z_1\otimes Z_2$. The ring $A_1\otimes A_2$ is simple (resp. semisimple) if and only if the ring $Z_1\otimes Z_2$ is a field (resp. a reduced ring).

This is the case $M_1= (A_1)_s, M_2= (A_2)_s$ of Proposition 5 of VIII, p. 219.