Exercise 14, § 5

Let A be a (commutative) ring, B an A-algebra such that the A-module B is faithful and of finite presentation. Now prove that the following conditions are equivalent: α) For every maximal ideal $\mathfrak{m}$ of A, the $A/\mathfrak{m}$-algebra $B/\mathfrak{m}B$ is central simple. β) B is an a-module projective, and if $B^0$ denotes the opposite algebra of B, the canonical homomorphism $B \otimes_A B^0 \to \mathrm{End}_A(B)$ is bijective. (Reduce to the case where A is a local ring, and apply Exercise 13.) One then says that B is an Azumaya algebra over A.