Identities involving additive maps on division rings
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908794817085440 |
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| author | Singh, Lovepreet Tiwari, S. K. |
| author_facet | Singh, Lovepreet Tiwari, S. K. |
| contents | Let $g$ be an additive map on a division ring $D$. In this paper, we study the functional identity $G_{1}(y)g(y)G_{2}(y) = H(y)$, where $G_{1}(Y), G_{2}(Y)$, $H(Y)$ are generalized polynomials in $D_{G}[Y]$ such that both $G_{1}(Y)$ and $G_{2}(Y)$ are non-zero. By application of this result and its implications, we prove that if $D$ is a non-commutative division ring with $\operatorname{char}(D) \neq 2$, then the only possible solution of additive maps $g_{1},g_{2}: D \rightarrow D$ satisfying the identity $g_{1}(y)y^{-m} + y^{n}g_{2}(y^{-1})= 0$ is $ g_{1} = g_{2} = 0$, where $m$ and $n$ are positive integers with $(m,n) \neq (1,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Identities involving additive maps on division rings Singh, Lovepreet Tiwari, S. K. Rings and Algebras 16K40, 16R50, 16R60 Let $g$ be an additive map on a division ring $D$. In this paper, we study the functional identity $G_{1}(y)g(y)G_{2}(y) = H(y)$, where $G_{1}(Y), G_{2}(Y)$, $H(Y)$ are generalized polynomials in $D_{G}[Y]$ such that both $G_{1}(Y)$ and $G_{2}(Y)$ are non-zero. By application of this result and its implications, we prove that if $D$ is a non-commutative division ring with $\operatorname{char}(D) \neq 2$, then the only possible solution of additive maps $g_{1},g_{2}: D \rightarrow D$ satisfying the identity $g_{1}(y)y^{-m} + y^{n}g_{2}(y^{-1})= 0$ is $ g_{1} = g_{2} = 0$, where $m$ and $n$ are positive integers with $(m,n) \neq (1,1)$. |
| title | Identities involving additive maps on division rings |
| topic | Rings and Algebras 16K40, 16R50, 16R60 |
| url | https://arxiv.org/abs/2412.19223 |