Dedekind domains: Overrings and semi-prime elements

This paper develops two themes: (1) the relation of the class group of a Dedekind domain A to that of an overring B and (2) the question of finding a nonzero, nonunit element x of a Dedekind domain A such that A/xA is regular.We obtain complete results in answer to the first question, giving a corollary concerning the realization of certain groups as class groups.We give various sufficient conditions in answer to the second question; some in terms of the class group, others concerning Dedekind domains which often arise in practice.In § 1 of the present paper, we study the class group of an overring B of a Dedekind domain A and determine its class group in terms of that of A. We generalize and also strengthen the results of § 1 of an earlier article [1].Combining several results, we obtain an interesting fact: if G is the class group of a Dedekind domain and G r is a homorphic image of G, then G r is the class group of a suitable Dedekind domain.Section 2 introduces the question of finding a nonunit x in a Dedekind domain A for which A/xA is a direct sum of fields.Although we obtain no definitive result, various sufficient conditions are given.These require in part the developments of § 1.We also give examples Dedekind domains with "pathological" class groups.I* We state two well known propositions which we will need by way of background.PROPOSITION 1.1.Let A be a Dedekind Domain with quotient field F. Let B be a ring such that icΰcί 7 .Then B -f] A p over those prime ideals P of A for which Bci p , PROPOSITION 1.2.Let A be a Dedekind domain with quotient field F. Let B be a ring such that A c B c F. Then B is a Dedekind domain.PROPOSITION 1.3.Let A be a Dedekind domain with quotient field F and let B be a ring such that AcβcF.The assignment I->IB is a homomorphism of the set of fractionary ideals of A onto the set of fractionary ideals of B.