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 a v x x ∈ X = o x x ∈ X for a ring R , where a v x 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 o 1 − e = a v e 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 .