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

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 X 3(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 X 3(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 X 3(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.