Jacobson radicals of Ore extensions

TL;DR

This paper characterizes the Jacobson radical of Ore extensions over certain rings with automorphisms and derivations, providing new formulas and conditions that partially answer an open question in ring theory.

Contribution

It introduces a formula for the Jacobson radical of Ore extensions with $q$-skew derivations over fields of characteristic zero and explores conditions under which the radical intersected with the base ring is nil.

Findings

The Jacobson radical of $R[x;\sigma, ext{D}]$ is given by $I igcap R + I_0$ under specified conditions.

$J(R[x;\sigma, ext{D}]) igcap R$ is nil if $\sigma$ is locally torsion and $R$ satisfies certain properties.

Provides partial answers to an open question by Greenfeld, Smoktunowicz, and Ziembowski.

Abstract

Let $R$ be a ring, $\sigma$ be an automorphism of $R$, and $D$ be a $\sigma$-derivation on $R$. We will show that if $R$ is an algebra over a field of characteristic $0$ and $D$ is $q$-skew, then $J(R[x;\sigma,D])=I\cap R+I_0$ where $I=\{r\in R : rx\in J(R[x;\sigma,D])\}$ and $I_0=\{\sum_{i\geq 1}r_ix^i: r_i\in I\}$. We will prove that $J(R[x;\sigma,D])\cap R$ is nil if $\sigma$ 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 $\sigma D=D\sigma$. This answers partially an open question by Greenfeld, Smoktunowicz and Ziembowski.