A note on r-precious ring

Abstract

An element of a ring [Formula: see text] is said to be [Formula: see text]-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 [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.