Exercise 15, § 5

Let B be an Azumaya A-algebra (Exercise 14). (a) Show that the centre of B is identical with the subring A of B and is a direct factor of the A-module B (cf. Chapter I, § 2, Exercise 21 and Chapter II, § 5, Exercise 4). (b) Show that, for every two-sided ideal $b$ of B, $b = (b \cap A)B$. (Note first that B is stable under every homomorphism of the A-module B and use the existence of a projector of B onto A.)