Existence of Clean Elements in A Matrix Ring Over ℤ

Abstract

A ring with unity is called clean, if every element is clean i.e. for every element there exist an idempotent element and a unit element such that . The set of matrices ( ) {[ ]| } is a ring with respect to the usual addition and multiplication operation of matrices. The ring ( ) is not clean for some integral domain , but the ring has eight forms of clean elements. Those clean elements are constructed by adding an idempotent element and a unit element. Since the ring ℤ of all integers is an integral domain, so the subring (ℤ) ( ) is not clean. In his paper, we discuss the existence of clean elements in (ℤ) based on the forms of clean elements in ( ) and proved the sufficiency and necessary condition of clean elements in (ℤ).