Differential poynomial rings over locally nilpotent rings need not be Jacobson radical

TL;DR

This paper demonstrates that differential polynomial rings over locally nilpotent rings do not necessarily possess the Jacobson radical property, providing counterexamples and specific conditions where the radical behavior differs.

Contribution

It answers a question by Shestakov by showing that R[X;D] need not be Jacobson radical even when R is locally nilpotent, expanding understanding of radical properties in differential polynomial rings.

Findings

R[X;D] need not be Jacobson radical over locally nilpotent rings.

J(R[X;D])∩R is a nil ideal when R is over an uncountable field and D is locally nilpotent.

Counterexamples to Jacobson radical behavior in differential polynomial rings.

Abstract

We answer a question by Shestakov on the Jacobson radical in differential polynomial rings. We show that if R is a locally nilpotent ring with a derivation D then R[X;D] need not be Jacobson radical. We also show that J(R[X;D])\cap R is a nil ideal of R in the case where D is a locally nilpotent derivation and R is an algebra over an uncountable field.