On the genus of reduced cozero-divisor graph of commutative rings

Abstract

Let R be a commutative ring with identity and let (x) be the principal ideal generated by \(x\in R.\) Let \(\varOmega (R)^*\) be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph \(\varGamma _r(R)\) of R is an undirected simple graph with \(\varOmega (R)^*\) as the vertex set and two distinct vertices (x) and (y) in \(\varOmega (R)^*\) are adjacent if and only if \((x)\nsubseteq (y)\) and \((y)\nsubseteq (x)\). In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.