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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.14249 |
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| _version_ | 1866913848698601472 |
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| author | Djuang, Felicia Servina Wijayanti, Indah Emilia Susanti, Yeni |
| author_facet | Djuang, Felicia Servina Wijayanti, Indah Emilia Susanti, Yeni |
| contents | Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the subgraph of $Cl(R)$ induced by the set $\{(e, u) : e \text{ is a nonzero idempotent element of } R\}$. In this study, we examine the structure of clean graphs over $\mathbb{Z}_{n}$ derived from their $Cl_2$ graphs and investigate their relationship with the structure of their idempotent graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14249 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on Z_n Djuang, Felicia Servina Wijayanti, Indah Emilia Susanti, Yeni Rings and Algebras 05C25, 05C60, 05C62, 05C76, 16U99 F.4.1 Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the subgraph of $Cl(R)$ induced by the set $\{(e, u) : e \text{ is a nonzero idempotent element of } R\}$. In this study, we examine the structure of clean graphs over $\mathbb{Z}_{n}$ derived from their $Cl_2$ graphs and investigate their relationship with the structure of their idempotent graphs. |
| title | Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on Z_n |
| topic | Rings and Algebras 05C25, 05C60, 05C62, 05C76, 16U99 F.4.1 |
| url | https://arxiv.org/abs/2505.14249 |