Weak clean index of a ring

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}}$\mathrm{.}Theweakcleanindexof$R$isdefinedas$\sup {|\chi(a)|: a\in R}$anditisdenotedby$\win®,$where$| \chi(a)| $ denotes the cardinality of the set $\chi (a)$. In this article, we characterize rings of weak clean indices $1$, $2$ and $3$.