On the genus of extended zero-divisor graph of commutative rings

Abstract

Let ${\mathcal {S}}$ be a commutative ring with $Z({\mathcal {S}})$ its set of zero-divisors. The extended zero-divisor graph of ${\mathcal {S}}$, denoted by $\Gamma '({\mathcal {S}})$, is an undirected graph with vertex set $Z^*({\mathcal {S}})$ and two distinct vertices $\delta$ and $\omega$ are adjacent if and only if $\delta {\mathcal {S}} \cap Ann(\omega ) \ne 0$ or $\omega {\mathcal {S}} \cap Ann(\delta ) \ne 0$. In this paper, we first characterize finite commutative rings whose extended zero-divisor graph is isomorphic to some well-known graphs and then we classify finite commutative rings whose extended zero-divisor graph is planar, toroidal or projective.