Derivations having divergence zero and closed polynomials over domains

Let R   be an integral domain. In this paper, we study the kernels of derivations of R[2]R[2] and closed polynomials in R[n]R[n]. Let D be an R  -derivation of R[2]R[2] having divergence zero. In the first half of this paper, when R   is a QQ-domain, we give a necessary and sufficient condition for KerD to be generated by one element over R  . In the second half, we give some sufficient conditions for a non-constant polynomial to be closed in R[n]R[n].