Rings in which every unit is a sum of a nilpotent and an idempotent

TL;DR

This paper explores generalizations of UU rings, focusing on rings where units are sums of nilpotents and idempotents, including cases with multiple commuting idempotents, expanding understanding of ring unit structures.

Contribution

It introduces and analyzes two new classes of rings where units are sums of nilpotents and idempotents, extending prior work on UU rings.

Findings

Characterization of rings where units are sums of nilpotent and idempotent elements.

Analysis of rings with units as sums of nilpotent and two commuting idempotents.

Extension of the concept of UU rings to broader classes of rings.

Abstract

A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.