On a $q$-Skew Amitsur's Theorem

TL;DR

This paper proves that under certain conditions, the Jacobson radical of a q-skew Ore extension over an uncountable field is nil, partially answering a 2019 question and extending Amitsur's theorem.

Contribution

It establishes the nilpotency of the constant part of the Jacobson radical in q-skew Ore extensions and confirms a version of Amitsur's theorem in characteristic zero.

Findings

The constant part of the Jacobson radical is nil.

The Jacobson radical equals a nil ideal times the Ore extension.

Partial answer to a 2019 open question.

Abstract

Let $R$ be an algebra over an uncountable field, $\sigma$ a locally torsion automorphism and $\delta$ a locally nilpotent left $\sigma$-derivation such that $q\sigma\delta = \delta\sigma$, where $q$ is a nonzero scalar. We show that the constant part of the Jacobson radical of the Ore extension $R[x;\sigma,\delta]$ 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 $R[x;\sigma, \delta]$ is $N[x;\sigma,\delta]$ for some nil ideal $N$ of $R$.