Exercise 15, § 5

Let $B$ be an Azumaya A-algebra (exer. 14). a) Now prove that the center of $B$ is identical with the subring $A$ of $B$ and is a direct factor of the a-module $B$ (cf. Chap. I, § 2, exer. 21, and Chap. II, § 5, exer. 4). b) Now prove that, for every two-sided ideal $b$ of B, one has $b = (b \cap A)B$. (First note that $b$ is stable under every endomorphism of the a-module B, and use the existence of a projector of B onto A.)