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 Algebra Discrete Math., 2018, Volume 26, Issue 1, Pages 130–143 (Mi adm676)  RESEARCH ARTICLE

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

Department of Mathematics, Saurashtra University, Rajkot, 360 005 India

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.

Keywords: quasilocal ring, special principal ideal ring, clique number of a graph, planar graph. Full text: PDF file (374 kB) References: PDF file   HTML file
MSC: 13A15, 05C25
Revised: 24.08.2018
Language:

Citation: P. Vadhel, S. Visweswaran, “Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case”, Algebra Discrete Math., 26:1 (2018), 130–143 Citation in format AMSBIB
\Bibitem{VadVis18} \by P.~Vadhel, S.~Visweswaran \paper Planarity of a spanning subgraph of the intersection graph of ideals of a~commutative ring~I, nonquasilocal case \jour Algebra Discrete Math. \yr 2018 \vol 26 \issue 1 \pages 130--143 \mathnet{http://mi.mathnet.ru/adm676} 

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