Corollary

Suppose that the algebras $A_1$ and $A_2$ are simple and that the center $Z_1$ of $A_1$ is reduced to K. Then the ring $A_1\otimes A_2$ has no other two-sided ideals than 0 and itself.

By assumption, we have $Z_1= K$, and since $A_2$ is simple, its center $Z_2$ is a field (VIII, p. 121, Corollary 1, a)). The ring $Z = Z_1\otimes Z_2$ is therefore a field, and the corollary follows from Proposition 3, a).