r-clean rings

TL;DR

This paper introduces the concept of r-clean rings, where each element is the sum of a regular and an idempotent, and explores their properties and relationship to clean rings.

Contribution

It defines r-clean rings, establishes their relation to clean rings, and investigates their properties, expanding the understanding of ring decompositions.

Findings

r-clean rings are a generalization of clean rings

Certain properties of r-clean rings are characterized

Relationships between r-clean and other ring classes are established

Abstract

An element of a ring R is called clean if it is the sum of an idempotent and a unit. A ring R is called clean if each of its element is clean. An element r \in R called regular if r = ryr for some y \in R. The ring R is regular if each of its element is regular. In this paper we define a ring is r-clean if each of its elements is the sum of a regular and an idempotent element. We give some relations between r-clean and clean rings. Finally we investigate some properties of r-clean rings.