On The Crosscap Of Generalized Zero-Divisor Graph Of Commutative Rings

Abstract

Let R be a commutative ring and Z(R)^∗ be the set of all nonzero zero- divisors. The generalized zero divisor graph of R is defined as the graph Γ_g(R) with vertex set Z(R)^∗ and two vertices x and y are adjacent if and only if ann_R(x) + ann_R(y) is an essential ideal of R. In this paper, we classify all finite commutative rings R for which Γ_g(R) has crosscap one