NOTE ON WEAKLY NIL CLEAN AND pi-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 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 2 R is said to be von Neumann
regular if x = x2y for some y 2 R, and x is said to be pi-regular if xn = x2ny for some y in R and some integer n ⩾ 1. It is proved that an element x in R is pi-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.