TITLE:
On Weak Nil Clean Rings
AUTHORS:
Zubayda M. Ibraheem, Norihan N. Fadil
KEYWORDS:
Clean Rings, Nil Clean Rings, Weak Nil Clean and Strongly π-Regular Rings
JOURNAL NAME:
Open Access Library Journal,
Vol.9 No.6,
June
29,
2022
ABSTRACT: If a ring R is called weak nil clean if every element in R can be expressed as the sum or difference of nilpotent element and idempotent, if further the idempotent element and nilpotent element commute the ring is called weak* nil clean. The purpose of this paper is to give some characterization and basic properties of weak nil clean rings. The main results of this work are: 1) Let R be a ring, then R is weak nil clean if and only if R/P(R) is weak nil clean; 2) In a commutative ring R, if x is weak nil clean element, then xm is a weak nil clean element if (x-y)m=∑k-0m (-1)2k (kn)xkym-k x,y∈R (2); 3) Let R be a ring with Idem(R) = {0,1}, then R is weak nil clean if and only if R is local ring and J(R) is Nil ideal.