Abstract
We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicative ly 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 Dadekind domain, S the set of all monic polynomials and T the set of all primitive polynomials of A[X], then A[X]<? and A[X]T are both Dadekind domains. We obtain the class groups of these new Dsdekind domains in terms of that of A.
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