Exercise 19, § 5

Let A be a ring, B and C two A-algebra and $\phi : B \to C$ an A-algebra homomorphism. Suppose that B is an Azumaya algebra (Exercise 14). Let $C'$ be the sub-A-algebra of C consisting of the elements which commute with the elements of $\phi(B)$. Show that C is isomorphic to the algebra $B \otimes_A C'$. (Consider C as a $B \otimes_A B^0$-module and apply Exercise 18, taking account of the fact that $B \otimes_A B^0$ is isomorphic to $\mathrm{End}_A(B)$.)