The existence of m-clean elements in a certain upper triangular matrix ring over integral domain ℤ.

Let R be a unital ring and m ≥ 2 be a positive integer. An element a ∈ R is called an m-potent element in R, if it satisfies a^m = a. A unital ring R is called m-clean if each of its elements can be written as the sum of an m-potent element and a unit element of R. This article aims to show the existence of m-clean elements in a certain upper triangular matrix ring over integral domain ℤ, by identifying m-potent and unit elements in the ring. So, m-clean elements can be constructed. In this article, we obtained two general forms of all m-potent and unit elements and therefore produced four general forms of all m-clean elements which generalizes some theorems of clean elements in a certain upper triangular matrix ring over integral domain ℤ. Furthermore, we provided some properties of m-cleanness under ring homomorphism. [ABSTRACT FROM AUTHOR]