On a question of G. Bergman and L. Small on Azumaya algebras

The following result related to a question of G. Bergman and L. Small [J. Algebra, 33 (1975), 435-462] is obtained:. Let R be an associative algebra over a field F, 1 ε{lunate} R, char F ≠ 2, 5, such that R satisfies the standard polynomial identity S_2n and for every maximal ideal M in R, R M doesn't satisfy all the identities of (n - 3) × (n - 3) matrices. Then R is Azumaya. We also obtain an improvement of a theorem of M. Artin and C. Procesi, we show:. Let R be a ring with a unit such that R satisfies S_2n and R M doesn't satisfy S_2k, k < n, for every maximal ideal M in R. Then R is Azumaya of constant rank n². We also show that to answer Bergman and Small's question affirmatively for given n, it suffices to check the non-vanishing of a polynomial S_2n^* on the matrix ring M_k(F), for k > n 2.