On $\mathbb{Z}G$-clean rings

Let $R$ be an associative ring with unity. An element $x \in R$ is called $\mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $\mathbb{Z}G$-regular element in $R$. A ring $R$ is called $\mathbb{Z}G$-clean if every element of $R$ is $\mathbb{Z}G$-clean. In this paper, we show that in an abelian $\mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $\frac{R}{Nil(R)}$ is $G$-regular. Furthermore, we characterize $\mathbb{Z}G$-clean rings. Also, this paper is involved with investigating $\mathbb{F}_{2}C_{2}$ as a social group and measuring influence a member of it’s rather than others.