On Weak JN-Clean Rings

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,...,R_n be rings. Then, R = Π_i=1ⁿ R_i is weak JN-clean if and only if each R_i for each i is weak JN-clean and at most one R_i is not nil clean. 3) Let R be weak JN-clean. Then, 2 ∈ J(R).