The existence of clean elements in a matrix ring over integral domain and its connections with g(x)-cleanness and strongly g(x)-cleanness

Abstract

An element a in a ring R with unity is called clean, if there exist an idempotent element e ∈ R and a unit element u ∈ R such that a = e + u. This article aims to show all of clean elements in a certain subring X3(R) of a matrix ring 3 × 3 over integral domain R and their connections with g(x)-cleanness and strongly g(x)-cleanness for some fixed polynomial g(x). To achieve it, we found out unit and idempotent elements in X3(R) for constructing clean elements and selected some fixed g(x) in the center of R for investigating their relations with g(x)-cleanness and strongly g(x)-cleanness. In this article, we obtained eight general forms of the clean elements in X3(R), g(x)-clean elements with g(x) = xn − x, which five forms of them were strongly g(x)-clean but the other three forms were not. The latter result was shown by providing counter examples.