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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.04266 |
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| _version_ | 1866911790783266816 |
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| author | Baloda, Barkha Kumar, Jitender |
| author_facet | Baloda, Barkha Kumar, Jitender |
| contents | Let $R$ be a ring with unity. The upper ideal relation graph $Γ_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if there exists a non-unit element $z \in R$ such that the ideals $(x)$ and $(y)$ contained in the ideal $(z)$. In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of $Γ_U(R)$, we determine all the non-local finite commutative rings $R$ whose upper ideal relation graph has genus at most $2$. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of $Γ_U(R)$ is either $1$ or $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper ideal relation graphs associated to rings Baloda, Barkha Kumar, Jitender Rings and Algebras Combinatorics 05C25 Let $R$ be a ring with unity. The upper ideal relation graph $Γ_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if there exists a non-unit element $z \in R$ such that the ideals $(x)$ and $(y)$ contained in the ideal $(z)$. In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of $Γ_U(R)$, we determine all the non-local finite commutative rings $R$ whose upper ideal relation graph has genus at most $2$. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of $Γ_U(R)$ is either $1$ or $2$. |
| title | Upper ideal relation graphs associated to rings |
| topic | Rings and Algebras Combinatorics 05C25 |
| url | https://arxiv.org/abs/2403.04266 |