Eigenvalues of the product matrices of finite commutative rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912852069056512 |
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| author | Dolžan, David |
| author_facet | Dolžan, David |
| contents | The product matrix of a finite commutative ring $R=\{x_1,x_2,\ldots,x_n\}$ and an element $u \in R$ is the matrix $A_u(R)=[a_{ij}]$, where $a_{ij}=1$ if $x_ix_j=u$, and $a_{ij}=0$ otherwise. This provides a natural extension of the concept of the adjacency matrix of the zero-divisor graph of a ring, which has been studied extensively. In this paper, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ of odd order and a unit $u$. By studying the structure of a finite local ring, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ and any $u \in R$ in two cases: when the Jacobson radical of $R$ has either the maximal or the minimal possible index of nilpotency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19301 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of the product matrices of finite commutative rings Dolžan, David Rings and Algebras Combinatorics The product matrix of a finite commutative ring $R=\{x_1,x_2,\ldots,x_n\}$ and an element $u \in R$ is the matrix $A_u(R)=[a_{ij}]$, where $a_{ij}=1$ if $x_ix_j=u$, and $a_{ij}=0$ otherwise. This provides a natural extension of the concept of the adjacency matrix of the zero-divisor graph of a ring, which has been studied extensively. In this paper, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ of odd order and a unit $u$. By studying the structure of a finite local ring, we find the characteristic polynomial of $A_u(R)$ for a local ring $R$ and any $u \in R$ in two cases: when the Jacobson radical of $R$ has either the maximal or the minimal possible index of nilpotency. |
| title | Eigenvalues of the product matrices of finite commutative rings |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2601.19301 |