Rings Such That $u-1$ Lies In $J^{\#}(R)$ For Each Unit $u$

TL;DR

This paper explores a new class of rings called $UJ^{ ext{#}}$ rings, where units are closely related to the ring's radical, extending known classes and analyzing their properties, constructions, and interactions with other ring types.

Contribution

It introduces and characterizes $UJ^{ ext{#}}$ rings, extending previous classes, and investigates their properties, constructions, and relations with other ring classes.

Findings

$UJ^{\#}$ rings generalize UU and UJ rings.

Characterizations of $UJ^{\#}$ rings in various contexts.

Conditions under which group rings are $UJ^{\#}$.

Abstract

We investigate the so-called {\it $UJ^{\#}$ rings}, a new type of rings in which every unit can be written as $1+j$ with $j\in J^{\#}(R)$. These rings were defined and studied by Saini-Udar in Czechoslovak Math. J. (2025) under the name {\it $\sqrt{J}U$ rings}. (See \cite{SU}.) This class extends both the classes of UU and UJ rings, but also has its own special properties. In this study, we present some additional results about $UJ^{\#}$ rings that supply those from \cite{SU} explaining their connections with Dedekind-finite, semi-potent and Boolean rings, respectively, as well as we give several characterizations in this direction. We also examine how these rings behave under common ring constructions and find conditions for group rings to be $UJ^{\#}$. Moreover, our establishments shed a clearer picture of how unit elements interact with radical-like parts of a ring.