Dedekind domains and rings of quotients

Luther Elic Claborn

Vol. 15 (1965), No. 1, 59–64

DOI: 10.2140/pjm.1965.15.59

Abstract

We study the relation of the ideal class group of a Dedekind domain $A$ to that of $A_S$, where $S$ is a multiplicatively closed subset of $A$. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an arbitrary Dedekind domain.

If $A$ is a Dedekind domain, $S$ the set of all monic polynomials and $T$ the set of all primitive polynomials of $A\left[X\right]$, then $A{\left[X\right]}_S$ and $A{\left[X\right]}_T$ are both Dedekind domains. We obtain the class groups of these new Dedekind domains in terms of that of $A$.

Mathematical Subject Classification

Primary: 13.20

Secondary: 13.80

Milestones

Received: 13 December 1963

Published: 1 March 1965