@article { author = {Farmani, Marzieh}, title = {On $\mathbb{Z}G$-clean rings}, journal = {Algebraic Structures and Their Applications}, volume = {8}, number = {1}, pages = {25-40}, year = {2021}, publisher = {Yazd University}, issn = {2382-9761}, eissn = {2423-3447}, doi = {10.22034/as.2020.1834}, abstract = {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.}, keywords = {Social group,Strongly ZG-regular,Von Neumann regular,ZG-clean,ZG-regular}, url = {https://as.yazd.ac.ir/article_1834.html}, eprint = {https://as.yazd.ac.ir/article_1834_67678c88da5df6a3fde922d7c1091103.pdf} }