Some results on the complement of a new graph associated to a commutative ring
The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. We denote the collection of all ideals of $R$ by $\mathbb{I}(R)$ and $\mathbb{I}(R)\backslash \{(0)\}$ by $\mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou,...
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Azarbaijan Shahide Madani University
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| Series: | Communications in Combinatorics and Optimization |
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doaj-8efaec97c4be4513926d47ca575573892020-11-24T22:23:56ZengAzarbaijan Shahide Madani UniversityCommunications in Combinatorics and Optimization 2538-21282538-21362017-06-012211913810.22049/CCO.2017.25908.1053Some results on the complement of a new graph associated to a commutative ringS. Visweswaran0A. Parmar1Department 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 one nonzero proper ideal. Let $R$ be a ring. We denote the collection of all ideals of $R$ by $\mathbb{I}(R)$ and $\mathbb{I}(R)\backslash \{(0)\}$ by $\mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, {\em A new graph associated to a commutative ring}, Discrete Math. Algorithm. Appl. {\bf 8} (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to $R$, denoted by $\Omega_{R}^{*}$ which is an undirected graph whose vertex set is $\mathbb{I}(R)^{*}\backslash \{R\}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if either $(Ann_{R}I)J = (0)$ or $(Ann_{R}J)I = (0)$. Several interesting theorems were proved on $\Omega_{R}^{*}$ in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of $\Omega_{R}^{*}$ and the ring-theoretic properties of $R$. The aim of this article is to investigate some properties of $(\Omega_{R}^{*})^{c}$, the complement of the new graph $\Omega_{R}^{*}$ associated to $R$.http://comb-opt.azaruniv.ac.ir/article_13644_2227.htmlAnnihilating ideal of a ringmaximal N-prime of $(0)$special principal ideal ringconnected graphdiametergirth |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
S. Visweswaran A. Parmar |
| spellingShingle |
S. Visweswaran A. Parmar Some results on the complement of a new graph associated to a commutative ring Communications in Combinatorics and Optimization Annihilating ideal of a ring maximal N-prime of $(0)$ special principal ideal ring connected graph diameter girth |
| author_facet |
S. Visweswaran A. Parmar |
| author_sort |
S. Visweswaran |
| title |
Some results on the complement of a new graph associated to a commutative ring |
| title_short |
Some results on the complement of a new graph associated to a commutative ring |
| title_full |
Some results on the complement of a new graph associated to a commutative ring |
| title_fullStr |
Some results on the complement of a new graph associated to a commutative ring |
| title_full_unstemmed |
Some results on the complement of a new graph associated to a commutative ring |
| title_sort |
some results on the complement of a new graph associated to a commutative ring |
| publisher |
Azarbaijan Shahide Madani University |
| series |
Communications in Combinatorics and Optimization |
| issn |
2538-2128 2538-2136 |
| publishDate |
2017-06-01 |
| description |
The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. We denote the collection of all ideals of $R$ by $\mathbb{I}(R)$ and $\mathbb{I}(R)\backslash \{(0)\}$ by $\mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, {\em A new graph associated to a commutative ring}, Discrete Math. Algorithm. Appl. {\bf 8} (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to $R$, denoted by $\Omega_{R}^{*}$ which is an undirected graph whose vertex set is $\mathbb{I}(R)^{*}\backslash \{R\}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if either $(Ann_{R}I)J = (0)$ or $(Ann_{R}J)I = (0)$. Several interesting theorems were proved on $\Omega_{R}^{*}$ in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of $\Omega_{R}^{*}$ and the ring-theoretic properties of $R$. The aim of this article is to investigate some properties of $(\Omega_{R}^{*})^{c}$, the complement of the new graph $\Omega_{R}^{*}$ associated to $R$. |
| topic |
Annihilating ideal of a ring maximal N-prime of $(0)$ special principal ideal ring connected graph diameter girth |
| url |
http://comb-opt.azaruniv.ac.ir/article_13644_2227.html |
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