### 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.

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