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 |
| Published: |
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 |
| Summary: | 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. |
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| ISSN: | 1319-5166 |