Exercise 7, § 12

Let $A_1$ and $A_2$ be semisimple K-algebras and $Z_1$ and $Z_2$ their centers. Suppose that $A_1$ has finite degree over K and that the ring $Z_1\otimes Z_2$ is reduced. Let $f$ be a K-algebra homomorphism from $A_2$ to $A_1$. Prove that the commutant of $f(A_2)$ in $A_1$ is a semisimple K-algebra (view $A_1$ as an $(A_2\otimes A^o_1$)-module and use Proposition 7 of VIII, p. 221).