On the Jacobson radical of skew polynomial extensions of rings satisfying a polynomial identity

TL;DR

This paper investigates the Jacobson radical of skew polynomial rings over rings satisfying polynomial identities, extending known results from commutative rings to more general PI rings.

Contribution

It generalizes the understanding of the Jacobson radical in skew polynomial extensions from commutative rings to rings satisfying polynomial identities.

Findings

$J(R[x;D])igcap R$ is a nil $D$-ideal

Extension of results from commutative to PI rings

Provides new insights into the structure of radicals in skew polynomial rings

Abstract

Let $R$ be a ring satisfying a polynomial identity and let $D$ be a derivation of $R$. We consider the Jacobson radical of the skew polynomial ring $R[x;D]$ with coefficients in $R$ and with respect to $D$, and show that $J(R[x;D])\cap R$ is a nil $D$-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when $R$ is commutative.