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arXiv · 2605.08100

On a $q$-Skew Amitsur's Theorem

Aristide F. J. -C. Launois
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Abstract

Let $R$ be an algebra over an uncountable field, $σ$ a locally torsion automorphism and $δ$ a locally nilpotent left $σ$-derivation such that $qσδ= δσ$, where $q$ is a nonzero scalar. We show that the constant part of the Jacobson radical of the Ore extension $R[x;σ,δ]$ is nil. This partially answers a question of Greenfeld, Smoktunowicz and Ziembowski posed in 2019. As a corollary, we employ Shin's 2024 result to prove a q-skew Amitsur's theorem whenever the field is additionally assumed to be of characteristic zero. That is, the Jacobson radical of $R[x;σ, δ]$ is $N[x;σ,δ]$ for some nil ideal $N$ of $R$.

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Aristide F. J. -C. Launois. 2026-04-22. On a $q$-Skew Amitsur's Theorem. https://arxiv.org/abs/2605.08100

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