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claborn65 - P ACIFIC J OURNAL O F M ATHEMATICS Vol. 15, No....

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PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 1. 1965 DEDEKIND DOMAINS AND RINGS OF QUOTIENTS LUTHER CLABORN 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 If A is a Dadekind domain, S the set of all monic poly- nomials 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 A. 1* LEMMA 1-1. If A is a Dedekind domain and S is a multi- plicatively closed set of A suoh that A s is not a field, then A s is also a Dedekind domain. Proof. That A s is integrally closed and Noetherian if A is, follows from the general theory of quotient ring formations. The primes of A s are of the type PA S) where P is a prime ideal of A such that PΓ)S = ψ. Since height PA S = height P if PΠS = φ, P Φ (0) and PΠS = φ imply that height PA S = 1. PROPOSITION 1-2. If A is a Dedekind domain and S is a multi- plicatively closed set of A, the assignment C » CA S is a mapping of the set of fractionary ideals of A onto the set of fractionary ideals A s which is a homomorphism for multiplication. Proof. C is a fractionary ideal of A if and only if there is a d G A dC S A. If this is so, certainly dCA s S A Sj so CA S is a fractionary ideal of A s . Clearly (J5 C)A S ~ BA S -CA S , so the assignment is a homomorphism. Let D be any fractionary ideal of A s . Since A s is a Dedekind domain, D is in the free group generated by all prime ideals of A s , i.e. D = Q - 1 Ql k . For each i — 1, , k there is a prime P i A Q i = P { A Sa Set E = Pi 1 P n k K Then using the fact that we have a multiplicative homomorphism of fractionary ideals, we get Received December 13, 1963. 59
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60 LUTHER CLABORN EA a = (P 1 A S )^ (P k A s y* = Q i . .. QlK COROLLARY 1-3. Let A be a Dedekind domain and S be a multi- plicatively closed set of A. Let C (for C a fractionary ideal of A or A s ) denote the class of the ideal class group to which C belongs. Then the assignment C * CA S is a homomorphism φ of the ideal class group of A onto that of A s . Proof It is only necessary to note that if C = dA, then CA S = dA s . THEOREM 1-4. The kernel of φ is generated by all P ω , where Pa ranges over all primes such that P#OS Φ Φ. If P a ΠS Φ φ, then P^As — A s . Suppose C is a fractionary ideal such that C = P ay i.e. C = dP ω for some d in the quotient field of A.
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claborn65 - P ACIFIC J OURNAL O F M ATHEMATICS Vol. 15, No....

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