Orbits of automorphisms of integral domains

Abstract

Let $R$ be an integral domain. We study the structure of $R$ under the condition that the orbit space $R/\mathit{Aut}(R)$ is finite. It is proved that if $R$ is Noetherian, then $ |R/\mathit{Aut}(R)|= \infty$ unless $R$ is a finite field (Theorem 15 and Corollary 16). Furthermore, we give an example of an infinite integral domain with $|R/\mathit{Aut}(R)|< \infty$.