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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2403.10161 |
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| _version_ | 1866914715499757568 |
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| author | Lande, Anita Khairnar, Anil |
| author_facet | Lande, Anita Khairnar, Anil |
| contents | Let $R$ be a ring with involution $*$ and $Z^*(R)$ denotes the set of all non-zero zero-divisors of $R$.
We associate a simple (undirected) graph $Γ'(R)$ with vertex set $Z^*(R)$ and two distinct vertices $x$ and $y$ are adjacent in $Γ'(R)$ if and only if $x^ny^*=0$ or $y^nx^*=0$, for some positive integer $n$. We find the diameter and girth of $Γ'(R)$. The characterizations are obtained for $*$-rings having $Γ'(R)$ a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring $R$, there is an involution on $R\times R$ such that $Γ'(R\times R)$ is disconnected if and only if $R$ is an integral domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10161 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized zero-divisor graph of $*$-rings Lande, Anita Khairnar, Anil Combinatorics Let $R$ be a ring with involution $*$ and $Z^*(R)$ denotes the set of all non-zero zero-divisors of $R$. We associate a simple (undirected) graph $Γ'(R)$ with vertex set $Z^*(R)$ and two distinct vertices $x$ and $y$ are adjacent in $Γ'(R)$ if and only if $x^ny^*=0$ or $y^nx^*=0$, for some positive integer $n$. We find the diameter and girth of $Γ'(R)$. The characterizations are obtained for $*$-rings having $Γ'(R)$ a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring $R$, there is an involution on $R\times R$ such that $Γ'(R\times R)$ is disconnected if and only if $R$ is an integral domain. |
| title | Generalized zero-divisor graph of $*$-rings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.10161 |