A symmetric generalization of $$\pi $$-regular 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 $$\pi $$ -regular rings (in particular, of the von Neumann regular rings and of the strongly $$\pi $$ -regular rings). Some other close relationships with certain well-known classes of rings such as 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 Linear Algebra Appl (2018). Likewise, the obtained symmetrization supports that similar property for exchange rings established by Khurana et al. in Algebras Represent Theory (2015).