Weakly g(x)-Clean Rings

A ring R with identity is called “clean” if for every element a ? R, there exist an idempotent e and a unit u in R such that a = u+e. Let C(R) denote the center of a ring R and g(x) be a polynomial in C(R)[x]. An element r ? R is called “g(x)-clean” if r = u + s where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. In this paper we define a ring to be weakly g(x)-clean if each element of R can be written as either the sum or difference of a unit and a root of g(x).