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 $ χ(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 $\sup \{|χ(a)|: a\in R\}$ and it is denoted by $\win(R),$ where $| χ(a)|$ denotes the cardinality of the set $χ(a)$. In this article, we characterize rings of weak clean indices $1$, $2$ and $3$.

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