On Semi-Nil Clean Rings with Applications

TL;DR

This paper introduces and studies semi-nil clean rings, exploring their properties, characterizations, and applications, including group rings, unit elements, and matrix rings, revealing new structural insights in ring theory.

Contribution

It defines semi-nil clean rings and related subclasses, providing characterizations and demonstrating their properties and behaviors in various algebraic constructions.

Findings

Semi-nil clean rings are periodic if NI.

Group rings of nilpotent groups over weakly 2-primal rings are semi-nil clean iff conditions hold.

Matrix rings over t-fine rings are also t-fine.

Abstract

We investigate the notion of \textit{semi-nil clean} rings, defined as those rings in which each element can be expressed as a sum of a periodic and a nilpotent element. Among our results, we show that if $R$ is a semi-nil clean NI ring, then $R$ is periodic. Additionally, we demonstrate that every group ring $RG$ of a nilpotent group $G$ over a weakly 2-primal ring $R$ is semi-nil clean if, and only if, $R$ is periodic and $G$ is locally finite. Moreover, we also study those rings in which every unit is a sum of a periodic and a nilpotent element, calling them \textit{unit semi-nil clean} rings. As a remarkable result, we show that if $R$ is an algebraic algebra over a field, then $R$ is unit semi-nil clean if, and only if, $R$ is periodic. Besides, we explore those rings in which non-zero elements are a sum of a torsion element and a nilpotent element, naming them \textit{t-fine} rings, which constitute a proper subclass of the class of all fine rings. One of the main results is that matrix rings over t-fine rings are again t-fine rings.