A Generalization of $\Delta$U Rings

TL;DR

This paper introduces and studies weakly ΔU-rings, characterizing their structure, relationships with classical ring concepts, and behavior under various extensions, expanding previous work on ΔU-rings.

Contribution

It defines weakly ΔU-rings, explores their properties, relationships with other ring classes, and characterizes when group rings are weakly ΔU, extending prior research.

Findings

Matrix rings are never WΔU for n ≥ 2.

Complete characterizations of local, semi-local, semi-simple, and semi-regular WΔU rings.

WΔU property in exchange rings is equivalent to WUJ.

Abstract

In this paper, we introduce and study a new class of rings calling them {\it weakly $\Delta U$-rings}, hereafter abbreviated as {\it $W\Delta U$-rings} for short. A ring $R$ is said to be $W\Delta U$ if every unit of $R$ can be expressed as $\pm 1 + d$ for some $d \in \Delta(R)$, where $\Delta(R)$ is the largest Jacobson radical of $R$ that is closed under multiplication by units. Utilizing the known structure of $\Delta(R)$, we investigate the relationships between $W\Delta U$ rings and certain classical concepts such as $\Delta U$-rings, $UJ$-rings, $WUJ$-rings, as well as clean and exchange rings. Among the main results, we show that a matrix ring $M_n(R)$ is never $W\Delta U$ for any $n \ge 2$. We also provide complete characterizations of local, semi-local, semi-simple and semi-regular rings that are $W\Delta U$. Furthermore, it is shown for exchange rings that the $W\Delta U$ property is equivalent to being $WUJ$. Furthermore, the behavior of $W\Delta U$-rings under various ring extensions, including skew polynomial rings, skew power series rings, triangular matrix rings, trivial extensions and group rings, is thoroughly examined. Several examples are given to illustrate that the class of $W\Delta U$-rings properly contains the class of $\Delta U$-rings. Finally, necessary and sufficient conditions for a group ring $RG$ to be $W\Delta U$ are established too. Resuming all of the presented above, our results expanded those by Karaba\c{c}ak et al. published in J. Algebra \& Appl. (2021).