A primeness property for central polynomials of verbally prime P.I. algebras

Abstract

Let and be two non-commutative polynomials in disjoint sets of variables. An algebra is verbally prime if whenever is an identity for then either or is also an identity. As an analogue of this property, Regev proved that the verbally prime algebra of matrices over an infinite field has the following primeness property for central polynomials: whenever the product is a central polynomial for then both and are central polynomials. In this paper, we prove that over a field of characteristic zero, Regev’ s result holds for the verbally prime algebras and , where is the infinite-dimensional Grassmann algebra.