Certain functional identities on division rings
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917613454491648 |
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| author | Lee, Tsiu-Kwen Lin, Jheng-Huei |
| author_facet | Lee, Tsiu-Kwen Lin, Jheng-Huei |
| contents | We study the functional identity $G(x)f(x)=H(x)$ on a division ring $D$, where $f \colon D\to D$ is an additive map and $G(X)\ne 0, H(X)$ are generalized polynomials in the variable $X$ with coefficients in $D$. Precisely, it is proved that either $D$ is finite-dimensional over its center or $f$ is an elementary operator. Applying the result and its consequences, we prove that if $D$ is a noncommutative division ring of characteristic not $2$, then the only solution of additive maps $f, g$ on $D$ satisfying the identity $f(x) = x^n g(x^{-1})$ with $n\ne 2$ a positive integer is the trivial case, that is, $f=0$ and $g=0$. This extends Catalano and Merchán's result in 2023 to get a complete solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03112 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Certain functional identities on division rings Lee, Tsiu-Kwen Lin, Jheng-Huei Rings and Algebras 16R60 (Primary) 16R50, 16K40 (Secondary) We study the functional identity $G(x)f(x)=H(x)$ on a division ring $D$, where $f \colon D\to D$ is an additive map and $G(X)\ne 0, H(X)$ are generalized polynomials in the variable $X$ with coefficients in $D$. Precisely, it is proved that either $D$ is finite-dimensional over its center or $f$ is an elementary operator. Applying the result and its consequences, we prove that if $D$ is a noncommutative division ring of characteristic not $2$, then the only solution of additive maps $f, g$ on $D$ satisfying the identity $f(x) = x^n g(x^{-1})$ with $n\ne 2$ a positive integer is the trivial case, that is, $f=0$ and $g=0$. This extends Catalano and Merchán's result in 2023 to get a complete solution. |
| title | Certain functional identities on division rings |
| topic | Rings and Algebras 16R60 (Primary) 16R50, 16K40 (Secondary) |
| url | https://arxiv.org/abs/2401.03112 |