TY - JOUR ID - 1834 TI - On $\mathbb{Z}G$-clean rings JO - Algebraic Structures and Their Applications JA - AS LA - en SN - 2382-9761 AU - Farmani, Marzieh AD - Islamic Azad university, Roudehen branch, Roudehen, Iran. Y1 - 2021 PY - 2021 VL - 8 IS - 1 SP - 25 EP - 40 KW - Social group KW - Strongly ZG-regular KW - Von Neumann regular KW - ZG-clean KW - ZG-regular DO - 10.22034/as.2020.1834 N2 - 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. UR - https://as.yazd.ac.ir/article_1834.html L1 - https://as.yazd.ac.ir/article_1834_67678c88da5df6a3fde922d7c1091103.pdf ER -