Generalizing unit-regular rings and special clean elements

Abstract

As a strengthening of the definition of weakly clean rings, given by Ster in J. Algebra (2014), and as a common generalization of the classical unit-regular rings, we define and investigate the class of socalled weakly unit-regular rings as those rings R for which, for every element a ? R, there exist a unit u and an idempotent e such that a ? u ? e ? (1 ? e)Ra with aR ? eR = {0}. Some more exotic relationships with the well-known classes of clean, nil-clean and (strongly) ?-regular rings are demonstrated as well. In particular, an elementwise extension of the so-called ”special clean elements” by Khurana et al. in J. Algebra & Appl. (2020) is also processed.