On the genus of nil-graph of ideals of commutative rings
Let R be a commutative ring with identity and let Nil(R) be the ideal of all nilpotent elements of R. Let I(R)={I:I is a non-trivial ideal of R and there exists a non-trivial ideal J such that IJ⊆Nil(R)}. The nil-graph of ideals of R is defined as the simple undirected graph AGN(R) whose vertex set...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Emerald Publishing
2017-07-01
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| Series: | Arab Journal of Mathematical Sciences |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S1319516616300342 |
| _version_ | 1819071526329122816 |
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| author | T. Tamizh Chelvam K. Selvakumar P. Subbulakshmi |
| author_facet | T. Tamizh Chelvam K. Selvakumar P. Subbulakshmi |
| author_sort | T. Tamizh Chelvam |
| collection | DOAJ |
| description | Let R be a commutative ring with identity and let Nil(R) be the ideal of all nilpotent elements of R. Let I(R)={I:I is a non-trivial ideal of R and there exists a non-trivial ideal J such that IJ⊆Nil(R)}. The nil-graph of ideals of R is defined as the simple undirected graph AGN(R) whose vertex set is I(R) and two distinct vertices I and J are adjacent if and only if IJ⊆ Nil(R). In this paper, we study the planarity and genus of AGN(R). In particular, we have characterized all commutative Artin rings R for which the genus of AGN(R) is either zero or one. |
| first_indexed | 2024-12-21T17:23:14Z |
| format | Article |
| id | doaj.art-0d6969cad32a4f71988960e6c0cb4dcf |
| institution | Directory Open Access Journal |
| issn | 1319-5166 |
| language | English |
| last_indexed | 2024-12-21T17:23:14Z |
| publishDate | 2017-07-01 |
| publisher | Emerald Publishing |
| record_format | Article |
| series | Arab Journal of Mathematical Sciences |
| spelling | doaj.art-0d6969cad32a4f71988960e6c0cb4dcf2022-12-21T18:56:07ZengEmerald PublishingArab Journal of Mathematical Sciences1319-51662017-07-0123218619510.1016/j.ajmsc.2016.09.004On the genus of nil-graph of ideals of commutative ringsT. Tamizh ChelvamK. SelvakumarP. SubbulakshmiLet R be a commutative ring with identity and let Nil(R) be the ideal of all nilpotent elements of R. Let I(R)={I:I is a non-trivial ideal of R and there exists a non-trivial ideal J such that IJ⊆Nil(R)}. The nil-graph of ideals of R is defined as the simple undirected graph AGN(R) whose vertex set is I(R) and two distinct vertices I and J are adjacent if and only if IJ⊆ Nil(R). In this paper, we study the planarity and genus of AGN(R). In particular, we have characterized all commutative Artin rings R for which the genus of AGN(R) is either zero or one.http://www.sciencedirect.com/science/article/pii/S1319516616300342Nil-graph of idealsCommutative ringAnnihilating-idealPlanarGenus |
| spellingShingle | T. Tamizh Chelvam K. Selvakumar P. Subbulakshmi On the genus of nil-graph of ideals of commutative rings Arab Journal of Mathematical Sciences Nil-graph of ideals Commutative ring Annihilating-ideal Planar Genus |
| title | On the genus of nil-graph of ideals of commutative rings |
| title_full | On the genus of nil-graph of ideals of commutative rings |
| title_fullStr | On the genus of nil-graph of ideals of commutative rings |
| title_full_unstemmed | On the genus of nil-graph of ideals of commutative rings |
| title_short | On the genus of nil-graph of ideals of commutative rings |
| title_sort | on the genus of nil graph of ideals of commutative rings |
| topic | Nil-graph of ideals Commutative ring Annihilating-ideal Planar Genus |
| url | http://www.sciencedirect.com/science/article/pii/S1319516616300342 |
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