Some results on a supergraph of the comaximal ideal graph of a commutative ring
The rings considered in this article are commutative with identity which admit at least two maximal ideals. We denote the set of all maximal ideals of a ring $R$ by $Max(R)$ and we denote the Jacobson radical of $R$ by $J(R)$. Let $R$ be a ring such that $|Max(R)|\geq 2$. Let $\mathbb{I}(R)$...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Azarbaijan Shahide Madani University
2018-01-01
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| Series: | Communications in Combinatorics and Optimization |
| Subjects: | |
| Online Access: | http://comb-opt.azaruniv.ac.ir/article_13778.html |
| _version_ | 1818759339455807488 |
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| author | S. Visweswaran J. Parejiya |
| author_facet | S. Visweswaran J. Parejiya |
| author_sort | S. Visweswaran |
| collection | DOAJ |
| description | The rings considered in this article are commutative with identity which admit at least two maximal ideals. We denote the set of all maximal ideals of a ring $R$ by $Max(R)$ and we denote the Jacobson radical of $R$ by $J(R)$. Let $R$ be a ring such that $|Max(R)|\geq 2$. Let $\mathbb{I}(R)$ denote the set of all proper ideals of $R$. In this article, we associate an undirected graph denoted by $\mathbb{INC}(R)$ with a subcollection of ideals of $R$ whose vertex set is $\{I\in \mathbb{I}(R)|I\not\subseteq J(R)\}$ and two distinct vertices $I_{1}, I_{2}$ are adjacent in $\mathbb{INC}(R)$ if and only if $I_{1}\not\subseteq I_{2}$ and $I_{2}\not\subseteq I_{1}$ (that is, $I_{1}$ and $I_{2}$ are not comparable under the inclusion relation). The aim of this article is to investigate the interplay between the graph-theoretic properties of $\mathbb{INC}(R)$ and the ring-theoretic properties of $R$. |
| first_indexed | 2024-12-18T06:41:09Z |
| format | Article |
| id | doaj.art-ff8321cd402a483fb3581b202b7986e8 |
| institution | Directory Open Access Journal |
| issn | 2538-2128 2538-2136 |
| language | English |
| last_indexed | 2024-12-18T06:41:09Z |
| publishDate | 2018-01-01 |
| publisher | Azarbaijan Shahide Madani University |
| record_format | Article |
| series | Communications in Combinatorics and Optimization |
| spelling | doaj.art-ff8321cd402a483fb3581b202b7986e82022-12-21T21:17:37ZengAzarbaijan Shahide Madani UniversityCommunications in Combinatorics and Optimization2538-21282538-21362018-01-013215117210.22049/CCO.2018.26132.1079Some results on a supergraph of the comaximal ideal graph of a commutative ringS. Visweswaran0J. Parejiya1Department of Mathematics, Saurashtra University, Rajkot, India 360 005Department of Mathematics, Saurashtra University, Rajkot, India 360 005The rings considered in this article are commutative with identity which admit at least two maximal ideals. We denote the set of all maximal ideals of a ring $R$ by $Max(R)$ and we denote the Jacobson radical of $R$ by $J(R)$. Let $R$ be a ring such that $|Max(R)|\geq 2$. Let $\mathbb{I}(R)$ denote the set of all proper ideals of $R$. In this article, we associate an undirected graph denoted by $\mathbb{INC}(R)$ with a subcollection of ideals of $R$ whose vertex set is $\{I\in \mathbb{I}(R)|I\not\subseteq J(R)\}$ and two distinct vertices $I_{1}, I_{2}$ are adjacent in $\mathbb{INC}(R)$ if and only if $I_{1}\not\subseteq I_{2}$ and $I_{2}\not\subseteq I_{1}$ (that is, $I_{1}$ and $I_{2}$ are not comparable under the inclusion relation). The aim of this article is to investigate the interplay between the graph-theoretic properties of $\mathbb{INC}(R)$ and the ring-theoretic properties of $R$.http://comb-opt.azaruniv.ac.ir/article_13778.htmlChained ringdiameter of a graphbipartite graphsplit graphcomplemented graph |
| spellingShingle | S. Visweswaran J. Parejiya Some results on a supergraph of the comaximal ideal graph of a commutative ring Communications in Combinatorics and Optimization Chained ring diameter of a graph bipartite graph split graph complemented graph |
| title | Some results on a supergraph of the comaximal ideal graph of a commutative ring |
| title_full | Some results on a supergraph of the comaximal ideal graph of a commutative ring |
| title_fullStr | Some results on a supergraph of the comaximal ideal graph of a commutative ring |
| title_full_unstemmed | Some results on a supergraph of the comaximal ideal graph of a commutative ring |
| title_short | Some results on a supergraph of the comaximal ideal graph of a commutative ring |
| title_sort | some results on a supergraph of the comaximal ideal graph of a commutative ring |
| topic | Chained ring diameter of a graph bipartite graph split graph complemented graph |
| url | http://comb-opt.azaruniv.ac.ir/article_13778.html |
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