GENERALIZING $\pi$-REGULAR RINGS

Abstract

We introduce the class of weakly nil clean rings, as rings $R$ in which for every $a \in R$ there existan 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 when the ring has central idempotents, or has bounded index of nilpotence, or is a PI-ring. Weakly nil clean rings also properly encompass nil clean rings of Diesl [13]. 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 [25], and the second author [33] to a wider class of ringsand provide partial answers to some open questions in [13] and [33]. Some other properties are studied and several examples are given as well.