Derivations and differential operators on rings and fields

TL;DR

This paper characterizes derivations of order n on rings of characteristic zero as limits of differential operators and shows compositions of nonzero derivations have exact order n.

Contribution

It establishes a topological characterization of derivations of order n and demonstrates that compositions of nonzero derivations attain the exact order n.

Findings

Derivations of order n are closures of differential operators of degree n in the product topology.

Compositions of nonzero derivations have exact order n.

Provides a topological framework for understanding derivations on rings.

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 $d_1, \dots, d_n$ are nonzero derivations on $R$, then $d_1 \circ \ldots \circ d_n$ is a derivation of exact order $n$.