Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case
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 inters...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1268 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-12682018-10-20T08:02:25Z Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case Vadhel, P. Visweswaran, S. quasilocal ring, special principal ideal ring, clique number of a graph, planar graph 13A15, 05C25 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. Lugansk National Taras Shevchenko University 2018-10-20 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1268 Algebra and Discrete Mathematics; Vol 26, No 1 (2018) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1268/700 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2018-10-20T08:02:25Z |
| collection |
OJS |
| language |
English |
| topic |
quasilocal ring special principal ideal ring clique number of a graph planar graph 13A15 05C25 |
| spellingShingle |
quasilocal ring special principal ideal ring clique number of a graph planar graph 13A15 05C25 Vadhel, P. Visweswaran, S. Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case |
| topic_facet |
quasilocal ring special principal ideal ring clique number of a graph planar graph 13A15 05C25 |
| format |
Article |
| author |
Vadhel, P. Visweswaran, S. |
| author_facet |
Vadhel, P. Visweswaran, S. |
| author_sort |
Vadhel, P. |
| title |
Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case |
| title_short |
Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case |
| title_full |
Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case |
| title_fullStr |
Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case |
| title_full_unstemmed |
Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case |
| title_sort |
planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring i, nonquasilocal case |
| description |
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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1268 |
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2025-07-17T10:36:09Z |
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