Strongly pure subgroups of separable torsion-free abelian groups

Abstract

In this paper we prove that countable strongly pure subgroups of completely decomposable groups are completely decomposable. We also show that strongly pure subgroups of separable torsion-free groups are separable. One of the questions that have been considered concerning completely decom- posable groups is the following: under what conditions is a subgroup of a com- pletely decomposable again completely decomposable? The well-known Baer- Kaplansky-Kulikov theorem asserts that direct summands of completely decompos- able groups are also completely decomposable. It is well known that pure subgroups of completely decomposable groups are not necessarily completely decomposable and, in 1972, L. Bican characterized all those completely decomposable groups any pure subgroup of which is completely decomposable. Recently, we have shown that homogeneous pure subgroups of a completely decomposable G are completely decomposable provided that the extractible typeset of G is countable. In this note we partially extend the Baer-Kaplansky-Kulikov theorem to strongly pure subgroups of completely decomposable groups. Specifically, we show that in a completely decomposable whose typeset satisfies the maximum condition all strongly pure subgroups are completely decomposable. We also show that countable strongly pure subgroups of completely decomposable groups are completely decom- posable. This not only extends but also gives a new proof of the classical theorem of Kulikov on countable direct summands of completely decomposable groups. Fi- nally, we shall show that strongly pure subgroups of separable torsion-free groups are separable, extending the well-known theorem of L. Fuchs on direct summands of separable groups. Throughout this note the word will mean an abelian group and for notation and terminology the standard reference is volume II of Fuchs's book (2). Let G be torsion-free and completely decomposable, i.e. G is a direct sum of rank one torsion-free groups. A type r is said to be an extractible type oi G it G has a rank one summand of type r. The extractible typeset of G, denoted by £(G), is the set of all extractible types of G. For every r € £{G), we have G(t) — GT © G*(t), where GT is a nonzero homogeneous completely decomposable of type r; GT is called a maximal t-homogeneous summand of G. If G = @t<=£(g) Gt is a decomposition of G such that GT is a maximal r-homogeneous summand for every t G £(G), then this decomposition is known as a homogeneous decomposition of G.