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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.11636 |
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| _version_ | 1866913693109846016 |
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| author | Stasinski, Alexander |
| author_facet | Stasinski, Alexander |
| contents | A well-known theorem of Fillmore says that if $A\in\operatorname{M}_{n}(K)$ is a non-scalar matrix over a field $K$ and $γ_{1},\dots,γ_{n}\in K$ are such that $γ_{1}+\dots+γ_{n}=\operatorname{Tr}(A)$, then $A$ is $K$-similar to a matrix with diagonal $(γ_{1},\dots,γ_{n})$. Building on work of Borobia, Tan extended this by proving that if $R$ is a unique factorisation domain with field of fractions $K$ and $A\in\operatorname{M}_{n}(R)$ is non-scalar, then $A$ is $K$-similar to a matrix in $\operatorname{M}_{n}(R)$ with diagonal $(γ_{1},\dots,γ_{n})$. We note that Tan's argument actually works when $R$ is any integrally closed domain and show that the result cannot be generalised further by giving an example of a matrix over a non-integrally closed domain for which the result fails. Moreover, Tan gave a necessary condition for $A\in\operatorname{M}_{n}(R)$ to be $R$-similar to a matrix with diagonal $(γ_{1},\dots,γ_{n})$. We show that when $R$ is a PID and $n\geq3$, Tan's condition is also sufficient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Fillmore's theorem over integrally closed domains Stasinski, Alexander Rings and Algebras A well-known theorem of Fillmore says that if $A\in\operatorname{M}_{n}(K)$ is a non-scalar matrix over a field $K$ and $γ_{1},\dots,γ_{n}\in K$ are such that $γ_{1}+\dots+γ_{n}=\operatorname{Tr}(A)$, then $A$ is $K$-similar to a matrix with diagonal $(γ_{1},\dots,γ_{n})$. Building on work of Borobia, Tan extended this by proving that if $R$ is a unique factorisation domain with field of fractions $K$ and $A\in\operatorname{M}_{n}(R)$ is non-scalar, then $A$ is $K$-similar to a matrix in $\operatorname{M}_{n}(R)$ with diagonal $(γ_{1},\dots,γ_{n})$. We note that Tan's argument actually works when $R$ is any integrally closed domain and show that the result cannot be generalised further by giving an example of a matrix over a non-integrally closed domain for which the result fails. Moreover, Tan gave a necessary condition for $A\in\operatorname{M}_{n}(R)$ to be $R$-similar to a matrix with diagonal $(γ_{1},\dots,γ_{n})$. We show that when $R$ is a PID and $n\geq3$, Tan's condition is also sufficient. |
| title | On Fillmore's theorem over integrally closed domains |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2502.11636 |