Vol. 15, No. 1, 1965

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Dedekind domains and rings of quotients

Luther Elic Claborn

Vol. 15 (1965), No. 1, 59–64

We study the relation of the ideal class group of a Dedekind domain A to that of AS, 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[X], then A[X]S and A[X]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
Received: 13 December 1963
Published: 1 March 1965
Luther Elic Claborn