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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.16342 |
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| _version_ | 1866909223485440000 |
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| author | Shin, Jooyoung |
| author_facet | Shin, Jooyoung |
| contents | Let $R$ be a ring, $σ$ be an automorphism of $R$, and $D$ be a $σ$-derivation on $R$. We will show that if $R$ is an algebra over a field of characteristic $0$ and $D$ is $q$-skew, then $J(R[x;σ,D])=I\cap R+I_0$ where $I=\{r\in R : rx\in J(R[x;σ,D])\}$ and $I_0=\{\sum_{i\geq 1}r_ix^i: r_i\in I\}$. We will prove that $J(R[x;σ,D])\cap R$ is nil if $σ$ is locally torsion and one of the following conditions is given: (1) $R$ is a PI-ring, (2) $R$ is an algebra over a field of characteristic $p>0$ and $D$ is a locally nilpotent derivation such that $σD=Dσ$. This answers partially an open question by Greenfeld, Smoktunowicz and Ziembowski. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16342 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Jacobson radicals of Ore extensions Shin, Jooyoung Rings and Algebras Let $R$ be a ring, $σ$ be an automorphism of $R$, and $D$ be a $σ$-derivation on $R$. We will show that if $R$ is an algebra over a field of characteristic $0$ and $D$ is $q$-skew, then $J(R[x;σ,D])=I\cap R+I_0$ where $I=\{r\in R : rx\in J(R[x;σ,D])\}$ and $I_0=\{\sum_{i\geq 1}r_ix^i: r_i\in I\}$. We will prove that $J(R[x;σ,D])\cap R$ is nil if $σ$ is locally torsion and one of the following conditions is given: (1) $R$ is a PI-ring, (2) $R$ is an algebra over a field of characteristic $p>0$ and $D$ is a locally nilpotent derivation such that $σD=Dσ$. This answers partially an open question by Greenfeld, Smoktunowicz and Ziembowski. |
| title | Jacobson radicals of Ore extensions |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2405.16342 |