Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case

Abstract

The rings considered in this article are nonzero  commutative with identity which 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 \(\mathbb{I}(R)^{*}\). Recall that the intersection graph of ideals of \(R\), denoted by \(G(R)\), is an undirected graph whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent if and only if \(I\cap J\neq (0)\). In this article, we consider a subgraph  of \(G(R)\), denoted by \(H(R)\), whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent in \(H(R)\) if and only if \(IJ\neq (0)\).  The purpose of this article is to characterize rings \(R\) with  at least two maximal ideals such that \(H(R)\) is planar.