Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929317775147008 |
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| author | Mathil, Praveen Baloda, Barkha Kumar, Jitender Somasundaram, A. |
| author_facet | Mathil, Praveen Baloda, Barkha Kumar, Jitender Somasundaram, A. |
| contents | Let $R$ be a commutative ring with unity. The prime ideal sum graph $\text{PIS}(R)$ of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings $R$ such that $\text{PIS}(R)$ is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_15335 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs Mathil, Praveen Baloda, Barkha Kumar, Jitender Somasundaram, A. Combinatorics Commutative Algebra 05C25 Let $R$ be a commutative ring with unity. The prime ideal sum graph $\text{PIS}(R)$ of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings $R$ such that $\text{PIS}(R)$ is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs. |
| title | Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs |
| topic | Combinatorics Commutative Algebra 05C25 |
| url | https://arxiv.org/abs/2210.15335 |