Geodesic
Planarity of a spanning subgraph of the intersection graph of ideals of a~commutative ring~I, nonquasilocal case
Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 130-143.

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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. 
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P. Vadhel; S. Visweswaran. Planarity of a spanning subgraph of the intersection graph of ideals of a~commutative ring~I, nonquasilocal case. Algebra and discrete mathematics, Tome 26 (2018) no. 1, pp. 130-143. http://geodesic.mathdoc.fr/item/ADM_2018_26_1_a12/

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