How far can we go with Amitsur's theorem?

TL;DR

This paper investigates the limits of Amitsur's theorem, demonstrating its failure for certain differential polynomial rings but confirming its validity under specific conditions involving locally nilpotent derivations in positive characteristic.

Contribution

It shows that Amitsur's theorem does not extend to all differential polynomial rings and identifies conditions under which it still holds.

Findings

Existence of a non-nil ring with a radical differential polynomial ring

Amitsur's theorem holds for locally nilpotent derivations in characteristic p>0

Counterexample for differential polynomial rings not satisfying these conditions

Abstract

A well-known theorem by S.A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x; D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x; D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p>0.