Exercise 19, § 5

Let $A$ be a ring, $B$ and $C$ two $A$-algebras, $\varphi : B \to C$ an homomorphism of $A$-algebras. We suppose that $B$ is an Azumaya algebra (exerc. 14). Let $C'$ be the sub-$A$-algebra of $C$ formed by the elements which commute with the elements of $\varphi(B)$. Now prove that $C$ is isomorphic to the algebra $B \otimes_A C'$. (Consider $C$ as a $B \otimes_A B^0$-module and apply exerc. 18 taking account of the fact that $B \otimes_A B^0$ is isomorphic to $\mathrm{End}_A(B)$.)