Rings additively generated by idempotents and nilpotents

TL;DR

This paper explores the structure of rings where elements can be expressed as sums of idempotents and nilpotents, revealing new connections between strongly 2-nil-clean rings and feebly clean rings, and relating them to weakly exchange rings.

Contribution

It establishes novel relationships between strongly 2-nil-clean rings and feebly clean rings, and links these to weakly exchange rings, expanding understanding of ring decompositions.

Findings

Strongly 2-nil-clean rings have specific structural properties.

Connections between strongly 2-nil-clean and feebly clean rings are established.

Relations to weakly exchange rings are demonstrated.

Abstract

A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly 2-nil-clean rings and weakly exchange rings