A Generalization Of Regular And Strongly $\Pi$-regular Rings

TL;DR

This paper introduces D-regularly nil clean rings, a new class that generalizes von Neumann regular and strongly π-regular rings, and explores their relationships with other well-known ring classes.

Contribution

The paper defines D-regularly nil clean rings and demonstrates their connections to various classical and modern ring classes, expanding the understanding of ring regularity concepts.

Findings

D-regularly nil clean rings generalize von Neumann regular rings.

They also extend strongly π-regular rings.

The work links these rings to exchange, clean, and nil-clean rings.

Abstract

We introduce and investigate the so-called D-regularly nil clean rings by showing that these rings are, in fact, a non-trivial generalization of the classical von Neumann regular rings and of the strongly $\pi$-regular rings. Some other close relationships with certain well-known classes of rings such as $\pi$-regular rings, exchange rings, clean rings, nil-clean rings, etc., are also demonstrated. These results somewhat supply a recent publication of the author in Turk. J. Math. (2019) as well as they somewhat expand the important role of the two examples of nil-clean rings obtained by Ster in Lin. Algebra \& Appl. (2018). Likewise, the obtained symmetrization supports that for exchange rings established by Khurana et al. in Algebras \& Represent. Theory (2015).