Torsion-free Abelian Groups with Precobalanced Finite Rank Pure Subgroups

Abstract

§1: Introduction. One of the best understood classes of torsion-free abelian groups is that of direct sums of subgroups of the rationals called the completely decomposable abelian groups. In 1965, M. C. R. Butler [Bu] introduced the investigation of pure subgroups of completely decomposable abelian groups of finite rank, which have come to be known as Butler groups. Bican in [B] showed that in the dass of finite rank torsion-free abelian groups, such a group B is a Butler group if and only if (*) whenever 0 → T → G → B → 0 is balanced exact and T is a torsion abelian group, then the sequence splits. Bican and Salce [BS] use (*) to define the class of Butler groups of arbitrary rank. There have been many authors investigating this class and more tractable subclasses. An interesting result in the countable rank case is that these are the groups in which all their pure subgroups of finite rank are in the class defined by Butler. This class is called the class of finitely Butler groups. Bican and Salce [BS] gave an example to show that there is an uncountable finitely Butler group that does not satisfy (*), while Arnold [A] has constructed an example of a countable finitely Butler group that is not a pure subgroup of a completely decomposable abelian group. Recently, Fuchs and Metelli [FM] have shown that in the countable case that the finitely Butler groups are also defined by the property that all their pure finite rank subgroups are prebalanced in the sense of Fuchs and Viljoen [FV]. Rangaswamy and the author [GR] have dualized this property which they call precobalanced. Thus it is natural to investigate torsion-free abelian groups in which all their pure finite rank subgroups are precobalanced. We let https://www.w3.org/1998/Math/MathML"> ℜ * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071761/1cf6e0c8-4c76-44ea-bec7-2cccf4e763f3/content/ieq0882.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be the collection of such groups. We are able to derive the following: