On ZG-clean rings

Let R be an associative ring with unity. An element x∈R is called ZG-clean if x=e+r, where e is an idempotent and r is a ZG-regular element in R. A ring R is called ZG-clean if every element of R is ZG-clean. In this paper, we show that in an abelian ZG-regular ring R, the Nil(R) is a two-sided ideal of R and RNil(R) is G-regular. Furthermore, we characterize ZG-clean rings. Also, this paper is involved with investigating F2C2 as a social group and measuring influence a member of it’s rather than others.