Group Action on the Set of Nonunits in Rings

Abstract

Let R be a ring, G be the group of all units of R, and X=R−G∪0. In this paper, we investigate avxx∈X=oxx∈X for a ring R, where avx is the set of all vertices of the zero-divisor graph of R adjacent to x. We also investigate the question on zero-divisor graphs posed in the literature such that when the equality o1−e=ave holds in a commutative regular ring R with identity. Here, e is a nonzero idempotent of R which is not the identity element of R.