Jacobson radicals of Ore extensions

Abstract

Let $R$ be a ring, $\sigma$ be an automorphism of $R$, and $D$ be a $\sigma$-derivation on $R$ with $D\sigma =\sigma D$. We will show that $J(R[x;\sigma ,D])\subseteq A[x;\sigma ,D]$ where $A$ is a collection of all leading coefficients of polynomials in $J(R[x;\sigma ,D])$. In addition, 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 and $D$ is $q$-skew, (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.

arXiv commentMost of Section 2 has been revised