On the Annihilator Graph of Group Rings

Abstract

Abstract. Let R be a commutative ring with nonzero identity and G be a nontrivial finite group. Also, let Z(R) be the set of zero-divisors of R and, for a ∈ Z(R), let ann(a) = {r ∈ R | ra = 0}. The annihilator graph of the group ring RG is defined as the graph AG(RG), whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices x and y are adjacent if and only if ann(xy) = ann(x) ∪ ann(y). In this paper, we study the annihilator graph associated to a group ring RG.

Annihilator graphs derived from group rings

“…And, for x ∈ Z * (RG), define ann(x) = {r ∈ RG/rx = 0}. Also for a given group ring RG and a finite subset X of the group G, we shall denote X the following element of RG: X = Also in 2017, Mojgan Afkhami defined the annihilator graph in group rings [9]. But here we give another definition of annihilator graphs in finite group rings and study their structures.…”