Torsion-free Modules over Valuation Domains Whose Balanced-Projective Dimension Is at Most One

Abstract

Let R denote a valuation domain. Torsion-free balanced-projective R-modules possess many interesting properties including a nice structure Ž w x. theory see 7 . In this paper, we study the larger class of torsion-free R-modules whose balanced-projective dimension is at most one. Fundamental to our investigation are the pure-essential submodules and a new concept of a pseudo-balanced submodule. The former helps to construct w x balanced extensions, as was done in 6, 9 , while the latter gives rise to a relative homological algebra. Starting from the proof of a result of the first w x author in 9 on the basic submodule of a completely decomposable torsion-free module, we find a noteworthy result: for a torsion-free R2 Ž . module B, if Bext B, T s 0 for all torsion R-modules T , then B must R have balanced-projective dimension F 1. Our results demonstrate how recent ideas from the theory of Butler groups can be successfully utilized in the study of modules over valuation domains. Specifically, the concept