Weak clean index of a ring

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 $3$.