Constants of derivations in polynomial rings over unique factorization domains

Abstract

A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any K {\mathcal K} -derivation of K [ x , y ] {\mathcal K}[x,y] , where K {\mathcal K} is a commutative field of characteristic zero, is a polynomial ring in one variable over K {\mathcal K} . In this paper we give an elementary proof of this theorem and show that it remains true if we replace K {\mathcal K} by any unique factorization domain of characteristic zero.