(n, g(x))-Clean Rings

A ring R is called n-clean ring if every element of R can be written as a sum of an idempotent and n units in R. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. An element is called g(x)-clean if r = u + s where u is a unit of R and g(s) = 0 and R is g(x)-clean if every element is g(x)-clean. An element x ∈ R is called (n, g(x))-clean if x = u1 + u2 + · · ·+ un + s, where g(s) = 0 and u1, u2,..., un are units in R and R is called (n, g(x))-clean ring if every element is (n, g(x))-clean. The class of clean rings, n-clean ring, and g(x)-clean rings is a proper subset of the class of (n, g(x))-clean rings. In this paper, we investigate some properties of (n, g(x))-clean rings