On the kernels of higher \(R\)-derivations of \(R[x_1, \ldots, x_n]\)

Abstract

Let \(R\) be an integral domain and \(A= R[x_1, \dots , x_n]\) be the polynomial ring in \(n\) variables. In this article, we study the kernel of higher \(R\)-derivation \(D\) of \(A\). It is shown that if \(R\) is a HCF ring and \(\operatorname{tr.deg}_R(A^D) \leq 1\) then \(A^D = R[f]\) for some \(f\in A\).