Exercise 10, § 14

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

a) Prove that the A-algebra S = End$_A(P)$ is an Azumaya algebra.

b) Prove that the left S-module P is projective.

c) Let $P'$ be a second faithful, projective, and finitely generated A-module. Prove that the A-module $P\otimes_AP'$ is faithful, projective, and finitely generated and that the algebra End$_A(P\otimes_AP')$ can be canonically identified with the algebra End$_A(P)\otimes_A$ End$_A(P')$.