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}_{۲}C_{۲} as a social group and measuring influence a member of it’s rather than others.

Keywords:

Von Neumann regular , ZG-regular , Strongly ZG-regular , ZG-clean , Social group