TITLE:
On Weak JN-Clean Rings
AUTHORS:
Zubayda M. Ibraheem, Norihan N. Fadil
KEYWORDS:
Clean Rings, Local Rings, Boolean Rings and Weak JN-Clean Rings
JOURNAL NAME:
Open Access Library Journal,
Vol.9 No.8,
August
17,
2022
ABSTRACT: We can say for a ring R weak JN-clean ring if all elements a in R it can be written as a sum or difference of nilpotent and idempotent. Further the nilpotent element belongs to the Jacobson radical of R. The purpose of this paper is to give some characterization and basic properties of this ring. Also we will studied the relationship between weak JN-clean rings and J-reduced ring, Boolean ring, local ring and clean ring: from the main result: 1) The ring R is weak JN-clean, if and only if, R/J(R) is weak JN-clean and each idempotent lifts modulo N(R). 2) Let R1,R2,R3,...,Rn be rings. Then, R = Πi=1n Ri is weak JN-clean if and only if each Ri for each i is weak JN-clean and at most one Ri is not nil clean. 3) Let R be weak JN-clean. Then, 2 ∈ J(R).