The decomposability of torsion free modules of finite rank

Abstract

Throughout this paper R will be an integral domain, not a field ; Q will be the quotient field of R; and K will be the i?-module QjR. R will be said to have property D, if every torsion-free i?-module of finite rank is a direct sum of /?-modules of rank one. In [9, §6] I attempted to give necessary and sufficient conditions for an integral domain to have property D. I have since discovered, however, that Lemma 6.4 of that paper is false, with the result that Theorem 6.1 is also false as stated, and needs some rather strong additional hypotheses to be resurrected. Other false lemmas based on Lemma 6.4 are Proposition 6.5 and Corollary 6.7; and Lemma 6.8 remains without a valid proof. Compounding the confusion, I have also discovered an error in Kaplansky's proof of his theorem [4] which purported to handle the Noetherian case, and this theorem remains without a valid proof. Proposition 6.11 and Corollary 6.12 of my paper [9] were based on Kaplansky's proof, and are false as stated. Corollary 6.13 remains without a valid proof. The purpose of this paper is to attempt to restore some order out of this chaos. In Theorem 4 I have settled the Noetherian case. This theorem states that a Noetherian domain has property D if and only if it is one of two types of Noetherian domains. However, I have not been able to find an example of a ring of type (1); i.e. a principal ideal domain R with exactly two nonzero prime ideals Px, P2 and such that RPl, RP2 are complete discrete valuation rings. It was precisely this type of ring that Kaplansky thought that he had proved could not exist. Thus one of the remaining unsolved tasks in finding all of the kinds of integral domains that have property D is either to produce an example of this type of ring, or to prove that it cannot exist. In Theorem 3 I have shown that if R is an /z-local domain with more than one maximal ideal, then R has property D if and only if R has exactly two maximal ideals Mx, M2 and RMl, RM2 are maximal valuation rings. The question of whether such a ring can exist is, of course, a generalization of the previous question raised. Theorem 2 proves that a valuation ring has property D if and only if it is a maximal valuation ring. The major part of this result, namely that a maximal valuation ring has property D, is originally due to Kaplansky [3, Theorem 12] who generalized Priifer's result [11] for complete discrete valuation rings. Both of