Generalizing pi-regular rings

TL;DR

This paper introduces weakly nil clean rings, a new class that generalizes pi-regular rings, and explores their properties, relationships with other ring classes, and implications for ring theory.

Contribution

The paper defines weakly nil clean rings, shows their relation to pi-regular rings, and extends existing results to this broader class, addressing open questions in ring theory.

Findings

Weakly nil clean rings are exchange rings.

They contain pi-regular rings as a proper subclass.

Every weakly nil clean ring of bounded index is strongly pi-regular.

Abstract

We introduce the class weakly nil clean rings, as rings R in which for every a\in R there exist an idempotent e and a nilpotent q such that a-e-q\in eRa. Every weakly nil clean ring is exchange. Weakly nil clean rings contain pi-regular rings as a proper subclass, and these two classes coincide in the case of central idempotents. Every weakly nil clean ring of bounded index and every weakly nil clean PI-ring is strongly pi-regular. The center of a weakly nil clean ring is strongly pi-regular, and consequently, every weakly nil clean ring is a corner of a clean ring. These results extend Azumaya [3], McCoy [24], and the second author [33] to a wider class of rings and provide partial answers to some open questions in [13] and [33]. Some other properties are also studied and several examples are given.