A note on r-precious ring

Abstract

An element of a ring R is said to be r-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring R are r-precious, then R is called an r-precious ring. We study some basic properties of r-precious rings. We also characterize von Neumann regular elements in M2(R) when R is a Euclidean domain and by this argument, we produce elements that are r-precious but either not r-clean or not precious. © 2021 World Scientific Publishing Company.