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

TL;DR

This paper extends Regev's primeness property for central polynomials from matrix algebras over infinite fields to certain verbally prime algebras involving the infinite dimensional Grassmann algebra, in characteristic zero.

Contribution

It proves that Regev's primeness property for central polynomials holds for $M_k(E)$ and $M_{k,k}(E)$ over fields of characteristic zero, expanding the class of algebras with this property.

Findings

Regev's primeness property holds for $M_k(E)$ and $M_{k,k}(E)$ in characteristic zero.

The result generalizes the primeness property from matrix algebras over infinite fields.

The proof involves properties of the infinite dimensional Grassmann algebra.

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.