Prime ideal graphs of commutative rings
<p style="text-align: justify;">Let <em>R</em> be a finite commutative ring with identity and <em>P</em> be a prime ideal of <em>R</em>. The vertex set is <em>R - </em>{0} and two distinct vertices are adjacent if their product in <e...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
InaCombS; Universitas Jember; dan Universitas Indonesia
2022-06-01
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| Series: | Indonesian Journal of Combinatorics |
| Subjects: | |
| Online Access: | http://www.ijc.or.id/index.php/ijc/article/view/156 |
| _version_ | 1797946207914426368 |
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| author | Haval Mohammed Salih Asaad A. Jund |
| author_facet | Haval Mohammed Salih Asaad A. Jund |
| author_sort | Haval Mohammed Salih |
| collection | DOAJ |
| description | <p style="text-align: justify;">Let <em>R</em> be a finite commutative ring with identity and <em>P</em> be a prime ideal of <em>R</em>. The vertex set is <em>R - </em>{0} and two distinct vertices are adjacent if their product in <em>P</em>. This graph is called the prime ideal graph of <em>R</em> and denoted by Γ<sub>P</sub>. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that Γ<sub>P</sub> is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of Γ<sub>P.</sub></p> |
| first_indexed | 2024-04-10T21:07:13Z |
| format | Article |
| id | doaj.art-1910b697e4df41e6adcbbaf4161b384c |
| institution | Directory Open Access Journal |
| issn | 2541-2205 |
| language | English |
| last_indexed | 2024-04-10T21:07:13Z |
| publishDate | 2022-06-01 |
| publisher | InaCombS; Universitas Jember; dan Universitas Indonesia |
| record_format | Article |
| series | Indonesian Journal of Combinatorics |
| spelling | doaj.art-1910b697e4df41e6adcbbaf4161b384c2023-01-22T03:10:03ZengInaCombS; Universitas Jember; dan Universitas IndonesiaIndonesian Journal of Combinatorics2541-22052022-06-0161424910.19184/ijc.2022.6.1.270Prime ideal graphs of commutative ringsHaval Mohammed Salih0Asaad A. Jund1Soran UniversitySoran University<p style="text-align: justify;">Let <em>R</em> be a finite commutative ring with identity and <em>P</em> be a prime ideal of <em>R</em>. The vertex set is <em>R - </em>{0} and two distinct vertices are adjacent if their product in <em>P</em>. This graph is called the prime ideal graph of <em>R</em> and denoted by Γ<sub>P</sub>. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that Γ<sub>P</sub> is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of Γ<sub>P.</sub></p>http://www.ijc.or.id/index.php/ijc/article/view/156prime ideal graph, nilpotent graph, clique number, chromatic number |
| spellingShingle | Haval Mohammed Salih Asaad A. Jund Prime ideal graphs of commutative rings Indonesian Journal of Combinatorics prime ideal graph, nilpotent graph, clique number, chromatic number |
| title | Prime ideal graphs of commutative rings |
| title_full | Prime ideal graphs of commutative rings |
| title_fullStr | Prime ideal graphs of commutative rings |
| title_full_unstemmed | Prime ideal graphs of commutative rings |
| title_short | Prime ideal graphs of commutative rings |
| title_sort | prime ideal graphs of commutative rings |
| topic | prime ideal graph, nilpotent graph, clique number, chromatic number |
| url | http://www.ijc.or.id/index.php/ijc/article/view/156 |
| work_keys_str_mv | AT havalmohammedsalih primeidealgraphsofcommutativerings AT asaadajund primeidealgraphsofcommutativerings |