Rings of nilpotent elements of monomial derivations on polynomial rings

We study the rings NilD of nilpotent elements of R-derivations D on polynomial rings over a UFD R containing Q. The main result of this paper determines NilD of the monomial derivations D on R[x,y]. Furthermore, as its application, we prove that, for each positive integer m, there exists a monomial R-derivation with a slice on R[x,y,z] such that the minimal number of generators of its kernel equals m.