Note on weakly nil clean and π-regular rings

Abstract

Let R be a commutative ring with identity 1 ? 0. The ring R is called weakly nil clean if every element x of R can be written as x = n + e or x = n ? e, where n is a nilpotent element of R and e is an idempotent element of R. The ring R is called weakly nil neat if every proper homomorphic image of R is weakly nil clean. Among other results, this paper gives some new characterizations of weakly nil clean (resp. weakly nil neat) rings. An element x ? R is said to be von Neumann regular if x = x2y for some y ? R, and x is said to be ?-regular if xn = x2ny for some y ? R and some integer n ? 1. It is proved that an element x ? R is ?-regular if and only if it can be written as x = n + r, where n is a nilpotent element and r is a von Neumann regular element. In this paper, we study the uniqueness of this expression.