Algebraic derivations on affine domains

TL;DR

This paper studies algebraic derivations on affine domains, exploring their graded ring structures and providing characterizations of polynomial rings in two variables through these derivations.

Contribution

It introduces new characterizations of polynomial rings in two variables using algebraic derivations on affine domains.

Findings

Algebraic derivations induce graded ring structures on affine domains.

Characterizations of polynomial rings in two variables are established.

Connections between derivations and the structure of affine domains are clarified.

Abstract

We observe algebraic derivations on an affine domain B defined over an algebraically closed field of characteristic 0, which are called locally finite derivations in commutative and non-commutative contexts in other references. We observe the graded ring structure which the algebraic derivation defines on B, and give characterizations of a polynomial ring in two variables in terms of algebraic derivations.