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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| Datum: | 2018 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | 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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Algebra and Discrete Mathematics| Zusammenfassung: | 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. |
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