On Weak Nil Clean Rings

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 x^m is a weak nil clean element if (x-y)^m=∑_k-0^m (-1)^2k (_kⁿ)x^ky^m-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.