On the genus of the annhilator 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 commutative ring \(R\) is the simple undirected graph \(\operatorname{AG}(R)\) with vertices \(Z(R)^*\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(\operatorname{ann}(xy)\neq \operatorname{ann}(x)\cup \operatorname{ann}(y)\). 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 \(\operatorname{AG}(R)\) has genus less or equal to one.