Exercise 17, § 5

Let $A$ be a ring, $P$ a finitely generated projective faithful a-module.

a) Now prove that the a-algebra $B = \mathrm{End}_A(P)$ is an Azumaya algebra. (First observe that the a-module B is isomorphic to $P \otimes_A P^*$, hence of finite presentation, then use § 2, No. 7, Prop. 19.)

b) Now prove that P is a projective left b-module (use Theorem 1 d) and § 3, Exercise 8 c)).

c) Let $P'$ be a second finitely generated projective faithful a-module. Now prove that $P \otimes_A P'$ is a finitely generated projective faithful a-module and that $\mathrm{End}_A(P \otimes_A P')$ is canonically identified with $\mathrm{End}_A(P) \otimes_A \mathrm{End}_A(P')$ (use Theorem 1 e), § 2, No. 7, Props. 18 and 19 and § 3, No. 3, Theorem 1). Case where $P'$ is of rank 1.