Every abelian group is a class group

Abstract

Let T be the set of minimal primes of a Krull domain A. If S is a subset of T9 we form B = n AP 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 AP 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\pesAp. It is immediate that B is also a Krull domain which contains A and has quotient field K. If one of the AP were eliminable from the intersection representing B, it would also be eliminable from that representing A. Thus the AP for PeS are exactly the rings of the type BQ, 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 BQ = AP. 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 AP are called the essential valuation rings, and we will denote by VP the valuation of A going with AP. 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 AP for PeS are the essential valuation rings of JB. Every ring B is of the form AM for some multiplicative set M if and only if the class group of A is torsion.