$\Sigma$-semicommutative rings and their skew PBW extensions

TL;DR

This paper introduces $\Sigma$-semicommutative rings, explores their properties and relationships with other ring classes, and studies how these properties influence skew PBW extensions, including Baer and quasi-Baer conditions.

Contribution

It defines $\Sigma$-semicommutative rings, relates them to other ring classes, and analyzes their impact on the structure and properties of skew PBW extensions.

Findings

$\Sigma$-semicommutative rings relate to Abelian, reduced, and other ring classes.

Under certain conditions, skew PBW extensions over these rings are Baer if and only if they are quasi-Baer.

Topological conditions for skew PBW extensions are also examined.

Abstract

In this paper, we introduce the concept of $\Sigma$-semicommutative ring, for $\Sigma$ a finite family of endomorphisms of a ring $R$. We relate this class of rings with other classes of rings such that Abelian, reduced, $\Sigma$-rigid, nil-reversible and rings satisfying the $\Sigma$-skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over $\Sigma$-semicommutative rings. We prove that if a ring $R$ is $\Sigma$-semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and only if $R$ is quasi-Baer, and equivalently, $A$ is quasi-Baer if and only if $A$ is Baer. Finally, we consider some topological conditions for skew PBW extensions over $\Sigma$-semicommutative rings.