On Artin's theorem and Azumaya algebras

A short proof of the existence of central polynomial of matrix rings is given and some of its applications. The following characterization of Azumaya algebras is proved: R is Azumaya iff there exist a_i, b_i ε{lunate} R, i = 1,..., k with ∑_i^k = 1 a_irb_i ε{lunate} Z(R), ∀r ε{lunate} R and ∑ a_ib_i = 1. This is proved as a consequence of the following generalization of a theorem due to M. Artin (and generalized by C. Procesi): Let R = Λ{x₁,..., x_k} be a p.i. ring, Λ a central noetherian subring. Then R is Azumaya iff for every two sided ideal I in R, Z( R I) = Z(R) I ∩ Z(R), where Z(R) denotes the center of R.