# Nice derivations over principal ideal domains

## TL;DR

This paper explores the extension of results on nice derivations from polynomial rings over fields to those over principal ideal domains, revealing that certain kernels are polynomial rings over the base field.

## Contribution

It extends the theory of nice derivations to polynomial rings over PIDs, showing kernels are polynomial rings under specific conditions.

## Key findings

- Kernel of a nice derivation in four variables over a field is a polynomial ring.
- Results generalize previous work from fields to principal ideal domains.
- Provides conditions under which derivation kernels are polynomial rings.

## Abstract

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on nice derivations of the polynomial ring in three variables over a field k of characteristic zero extend to the polynomial ring over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on the polynomial ring in four variables over k of rank at most three is a polynomial ring over k.