A Criterion for Complete Decomposability and Butler Modules over Valuation Domains

Abstract

Throughout this paper R will denote a valuation domain. P. Eklof and w x L. Fuchs 3 showed that an R-module M is free if and only M is a Baer 1 Ž . module, that is, Ext M, T s 0 for all torsion R-modules T. What can we R 1 Ž . say about R-modules M for which Bext M, T s 0 for all torsion modR ules T ? Here Bext is a subfunctor of Ext consisting of balanced R R extensions. Torsion-free R-modules M with the last property are called Butler modules. Is a Butler R-module M completely decomposable? L. w x Fuchs and E. Monari-Martinez 7 answered this in the affirmative when rank M F / . This paper attempts to investigate this question using the 1 w x recent work 11 on modules with balanced-projective dimension one. Ž . It is known that every torsion-free R-module M possesses a G / 9-family 0 Ž . C of pure submodules, namely, C satisfies the following conditions: 1 0, Ž . Ž . M g C , 2 C is closed under unions of chains, and 3 for any A g C and any countable subset X of M, there is a B g C such that A j X ; B and BrA has countable rank. For instance, we can take C to consist of all the Ž . pure submodules of M. If the condition 2 above is replaced by the Ž . condition 2 9 S N g C if N g C for each i g I, then we call C an ig I i i axiom-39 family. These are modifications of the original definition of an Ž w x. axiom-3 family by Paul Hill see, e.g., 1 and have also occurred with w x other variations under the name tight systems in 8 . Ž . Our main theorem Theorem 4.6 asserts that a Butler R-module B will be completely decomposable if and only if B has an axiom-39 family of pure submodules. We first show that the above condition on B is equiva-