Crosscap of the ideal based zero-divisor graph

Abstract

Let R be a commutative ring and I be an ideal of R. The ideal based zero-divisor graph, denoted by ΓI(R), is the graph with the vertex set {x∈R−I:xy∈Ifor somey∈R−I} and two distinct vertices x and y are adjacent if and only if xy∈I. In this paper, we classify all finite quotient rings R/I and ideals I of R for which the crosscap of ΓI(R) is at most one. Moreover, we investigate certain properties on the crosscap of ΓI(R) in the general case also.

Keywords

• Zero-divisor graph
• Ideal based zero-divisor graph
• Local ring
• Crosscap of a graph