Derivations and differential operators on rings and fields

Abstract. Let \(R\) be an integral domain of characteristic zero. We prove that a function \(D\colon R\to R\) is a derivation of order \(n\) if and only if \(D\) belongs to the closure of the set of differential operators of degree \(n\) in the product topology of \(R^R\), where the image space is endowed with the discrete topology. In other words, \(f\) is a derivation of order \(n\) if and only if, for every finite set \(F \subset R\), there is a differential operator \(D\) of degree \(n\) such that \(f=D\) on \(F\). We also prove that if \(d_1 ,\ldots , d_n\) are nonzero derivations on \(R\), then \(d_1 \circ \ldots \circ d_n\) is a derivation of exact order \(n\).