Primeness property for central polynomials of verbally prime P.I. algebras

Abstract

Let $f$ and $g$ be two noncommutative polynomials in disjoint sets of variables. An algebra $A$ is verbally prime if whenever $f\cdot g$ is an identity for $A$ then either $f$ or $g$ is also an identity. As an analogue of this property Regev proved that the verbally prime algebra $M_k(F)$ of $k\times k$ matrices over an infinite field $F$ has the following primeness property for central polynomials: whenever the product $f\cdot g$ is a central polynomial for $M_k(F)$ then both $f$ and $g$ are central polynomials. In this paper we prove that over a field of characteristic zero Regev' s result holds for the verbally prime algebras $M_k(E)$ and $M_{k,k}(E)$, where $E$ is the infinite dimensional Grassmann algebra.