Annihilator graphs derived from group rings

Abstract

Let RG be the group ring of the group G over a ring R and let Z∗(RG) be the collection of all non-zero zero divisors in a finite group ring RG. And, for x ∈ Z∗(RG), ann(x) = { r ∈ RG | rx = 0}. The annihilator graph of RG denoted as AG(RG), and is defined as the graph whose vertex set is the elements of non-zero zero divisors in RG and two distinct vertices vx and vy are adjacent if and only if ann(x) ∩ ann(y) ≠ 0. In this paper we try to characterize the properties of annihilator graphs in group rings.