On the genus of the annihilator graph of a commutative ring

Abstract

Let R be a commutative ring and Z(R) ∗ be its

set of non-zero zero-divisors. The annihilator graph of a commu- tative ring R is the simple undirected graph AG(R) with vertices

Z(R) ∗ , and two distinct vertices x and y are adjacent if and only if ann(xy) 6= ann(x) ∪ ann(y). The notion of annihilator graph has been introduced and studied by A. Badawi [7]. In this paper, we determine isomorphism classes of finite commutative rings with identity whose AG(R) has genus less or equal to one.