Rings involving idempotents, units and nilpotent elements

TL;DR

This paper introduces new classes of rings called weakly precious and precious rings, generalizing existing concepts, and explores their fundamental properties and subclasses.

Contribution

It defines and studies the properties of weakly precious and precious rings, extending the theory of clean and nil-clean rings with new subclasses.

Findings

Introduction of weakly precious and precious rings

Identification of fundamental properties of these rings

Development of new subclasses and types of weakly clean and nil-clean rings

Abstract

We define the concepts of weakly precious and precious rings which generalize the notions of weakly clean and nil-clean rings. We obtain some fundamental properties of these rings. We also obtain certain subclasses of such rings and then offer new kinds of weakly clean rings and nil-clean rings.