Sums of commuting potent and nilpotent elements in rings

Abstract

There has been a great deal of interest lately in studying rings in which every element can be written as a sum of potent elements and nilpotent elements. In this paper, we focus primarily on rings in which every element can be written as the sum of some number of [Formula: see text]-potent elements and a nilpotent (for some choice of [Formula: see text]), all of which commute with each other. The research that has been conducted thus far in this area has revealed a number of interesting results and intriguing patterns. Our goal in this paper is to unify and extend these results, for the purpose of organizing the current knowledge and facilitating future research in this area.