Dedekind domains and graded rings

Fabien Decruyenaere, Eric Jespers
1990 Proceedings of the American Mathematical Society  
We prove that a Dedekind domain R , graded by a nontrivial torsionfree abelian group, is either a twisted group ring k'[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.
doi:10.1090/s0002-9939-1990-1027092-7 fatcat:4yvxxkibfzgvljdd7qdwqtvp6a