Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring k[X,Y,Z] over a field k of characteristic zero extend to the polynomial ring R[X,Y,Z] over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on k[X1,X2,X3,X4] of rank at most three is a polynomial ring over k.