Dedekind domains and graded rings

Abstract:

We prove that a Dedekind domain $R$, graded by a nontrivial torsionfree abelian group, is either a twisted group ring ${k^t}[G]$ or a polynomial ring $k[X]$, where $k$ is a field and $G$ is an abelian torsionfree rank one group. It follows that $R$ is a Dedekind domain if and only if $R$ is a principal ideal domain. We also investigate the case when $R$ is graded by an arbitrary nontrivial torsionfree monoid.