Primeness property for graded central polynomials of verbally prime algebras

TL;DR

This paper investigates the primeness property for graded central polynomials in matrix algebras, identifying specific gradings that satisfy this property and extending results to certain algebraic structures.

Contribution

It characterizes which graded structures of matrix algebras have the primeness property for central polynomials and extends the property to broader classes of algebras.

Findings

Crossed product gradings satisfy the primeness property.

Sufficient conditions are provided for $M_n(R)$ to have the primeness property.

Algebras $M_{a,b}(E)$ satisfy the primeness property for ordinary central polynomials.

Abstract

Let $F$ be an infinite field. The primeness property for central polynomials of $M_n(F)$ was proved by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where $R$ admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions - satisfied by $M_n(E)$ with the trivial grading - to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence over a field of characteristic zero every verbally prime algebra as the primeness property.