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claborn66 - P ACIFIC J OURNAL O F M ATHEMATICS Vol 18 No 2...

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PACIFIC JOURNAL OF MATHEMATICS Vol. 18, No. 2, 1966 EVERY ABELIAN GROUP IS A CLASS GROUP LUTHER CLABORN Let T be the set of minimal primes of a Krull domain A. If S is a subset of T 9 we form B = n A P for PeS and study the relation of the class group of B to that of A. We find that the class group of B is always a homomorphic image of that of A. We use this type of construction to obtain a Krull domain with specified class group and then alter such a Krull domain to obtain a Dedekind domain with the same class group. Let A be a Krull domain with quotient field K. Thus A is an intersection of rank 1 discrete valuation rings; and if x e K, x is a unit in all but a finite number of these valuation rings. If P is a minimal prime ideal of A, then A P is a rank 1 discrete valuation ring and must occur in any intersection displaying A as a Krull domain. In fact, if T denotes the set of minimal prime ideals of A, then A = OPΘΓAP displays A as a Krull domain. Choose a subset S of T (S Φ 0) and form the domain B — Γ\pesA p . It is immediate that B is also a Krull domain which contains A and has quotient field K. If one of the A P were eliminable from the intersection representing B, it would also be eliminable from that representing A. Thus the A P for PeS are exactly the rings of the type B Q , where Q is a minimal prime ideal of B. If Q is minimal prime ideal of B, then Q Π A = P for the Pe S such that B Q = A P . Let A and B be generic labels throughout this paper for a Krull domain A and a Krull domain B formed from A as above. We recall that the valuation rings A P are called the essential valuation rings, and we will denote by V P the valuation of A going with A P . We summarize and add a complement to the above. PROPOSITION 1. With A and B as above, B is a Krull domain containing A, and the A P for PeS are the essential valuation rings of JB. Every ring B is of the form A M for some multiplicative set M if and only if the class group of A is torsion. Proof. Everything in the first assertion has been given above.
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