Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

TL;DR

This paper introduces the concept of weak nil cleanness in rings, generalizing previous notions, and provides comprehensive characterizations and decomposition theorems for such rings, including abelian rings and matrix rings over division rings.

Contribution

It defines weak nil cleanness, extends the theory of nil cleanness, and characterizes weakly nil-clean rings in various algebraic contexts.

Findings

Decomposition theorem for weakly nil-clean rings

Complete characterization of weakly nil-clean abelian rings

Determination of when matrix rings over division rings are weakly nil-clean

Abstract

{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it weak nil cleanness} for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is {\it weakly nil-clean}.} It is also completely determined when a matrix ring over a division ring is weakly nil-clean.