Exercise 14, § 5

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