A note on ext-completions

Abstract

1. Harrison [l] has introduced the notion of the ext-completion of an abelian group as follows. Let A be an abelian group. The ext-completion, A*, of A is defined to be A* = Ext(Q/Z, A). Subsequently Rotman [4] has made the topological analogy precise by furnishing abelian groups with a (not necessarily Hausdorff) topology in which ext-completions are closed whenever they are imbedded as subspaces. Megibben [3] has shown how the ext-completion may be used as an aid in the classification of mixed groups of torsion-free rank one. A result which is implicit in Megibben’s work is that every reduced non-splitting abelian group, with torsion-free rank one, and torsion part T, may be imbedded in T”. Following Harrison we call a group adjusted if it has no torsion-free direct summands. The above result has moved both Rotman and Hirsch* to ask the following question. Can every reduced adjusted group be imbedded in the ext-completion of its torsion part. 7 It turns out that the answer to this question must be in the negative.