Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring II, Quasilocal Case

Abstract

The rings we consider in this article are commutative with identity \(1\neq 0\) and are not fields. Let \(R\) be a ring. We denote the collection of all proper ideals of \(R\) by \(\mathbb{I}(R)\) and the collection \(\mathbb{I}(R)\setminus \{(0)\}\) by \(\mathbb{I}(R)^{*}\). Let \(H(R)\) be the graph associated with \(R\) whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent if and only if \(IJ\neq (0)\). The aim of this article is to discuss the planarity of \(H(R)\) in the case when \(R\) is quasilocal.