On Rings with the 2-UNJ Property

TL;DR

This paper introduces 2-UNJ rings, a new class generalizing existing rings, and explores their properties, relationships with other ring classes, and conditions under which they appear in matrix and group rings.

Contribution

The paper defines 2-UNJ rings, shows their relation to known classes, and provides new characterizations and examples involving matrix rings and group rings.

Findings

Every 2-UJ, 2-UU, or UNJ ring is 2-UNJ.

Counter-examples show the converse does not hold.

Conditions for group rings to be 2-UNJ are established.

Abstract

In this paper, we introduce a new class of rings calling them {\it 2-UNJ rings}, which generalize the well-known 2-UJ, 2-UU and UNJ rings. Specifically, a ring $R$ is called 2-UNJ if, for every unit $u$ of $R$, the inclusion $u^2 \in 1 + Nil(R) + J(R)$ holds, where $Nil(R)$ is the set of nilpotent elements and $J(R)$ is the Jacobson radical of $R$. We show that every 2-UJ, 2-UU or UNJ ring is 2-UNJ, but the converse does {\it not} necessarily hold, and we also provide counter-examples to demonstrate this explicitly. We, moreover, investigate the connections between these rings and other algebraic properties such as being potent, tripotent, regular and exchange rings, respectively. In particular, we thoroughly study some natural extensions, like matrix rings and Morita contexts, obtaining new characterizations that were not addressed in previous works. Furthermore, we establish conditions under which group rings satisfy the 2-UNJ property. These results not only provide a better understanding of the structure of 2-UNJ rings, but also pave the way for future intensive research in this area.