Weak clean index of a ring

TL;DR

This paper introduces the weak clean index of rings, a new measure based on elements' properties, and characterizes rings with weak clean indices 1, 2, and 3.

Contribution

It defines the weak clean index of rings and provides characterizations for rings with weak clean indices 1, 2, and 3.

Findings

Characterization of rings with weak clean index 1

Characterization of rings with weak clean index 2

Characterization of rings with weak clean index 3

Abstract

Motivated by the concept of clean index of rings, we introduce the concept of weak clean index of rings. For any element $a$ of a ring $R$ with unity, we define $ \chi(a)=\{e\in R\mid e^2=e\text{ and }a-e \mbox{ or } a+e \mbox{ is a unit}\}$. The weak clean index of $R$ is defined as $\sup \{|\chi(a)|: a\in R\}$ and it is denoted by $\win(R),$ where $| \chi(a)| $ denotes the cardinality of the set $\chi(a)$. In this article, we characterize rings of weak clean indices $1$, $2$ and $3$.