Azumaya-like properties of verbally prime algebras

Abstract

is a p.i. for A if and only if .f or g is. He proves over a field F of characteristic zero that a verbally prime algebra must be p.i. equivalent to one of the following: (1) 0, the trivial case, or (2) M,(F), II x n matrices over the field F, or (3) M,(E), II x II matrices over an infinite dimensional Grassmann algebra E, or (4) M,,,> which we will not define, or (5) the free algebra, a case we shall ignore by always taking A to be pi. Since [3], a number of papers have appeared in which pi. properties of n x n matrices have been generalized to verbally prime algebras. One such paper is [4], in which Razmyslov studies the trace identities of Mk,) and the central polynomials of Mk,, and M,(E). The present paper grew out of an attempt to generalize the Artin-Procesi theorem (cf. [a]) to the verbally prime case. We found that all verbally prime p.i. algebras satisfy weaker versions of the basic properties of Azumaya algebras. We will show as straightforward applications of Kemer’s and Razmyslov’s results