On differential polynomial rings over locally nilpotent rings

TL;DR

This paper proves that differential polynomial rings over locally nilpotent rings cannot be mapped onto rings with non-zero idempotents, addressing a recent open question in ring theory.

Contribution

It establishes a new property of differential polynomial rings over locally nilpotent rings, answering a specific open problem in the field.

Findings

Differential polynomial rings over locally nilpotent rings cannot map onto rings with non-zero idempotents.

The result provides insight into the structure of such rings and their limitations.

Addresses a recent question by Greenfeld, Smoktunowicz, and Ziembowski.

Abstract

Let $\delta$ be a derivation of a locally nilpotent ring $R$. Then the differential polynomial ring $R[X; \delta]$ cannot be mapped onto a ring with a non-zero idempotent. This answers a recent question by Greenfeld, Smoktunowicz and Ziembowski.