Homomorphic images of completely decomposable finite rank torsion-free groups (Q1176646)

Let \(A=A_ 1\oplus\dots\oplus A_ n\) be an abelian group which is the direct sum of rank-1-groups. In this paper necessary and sufficient conditions are given such that every torsion free image of \(A\) is again completely decomposable. There is an algebraic weakening, namely that \(A/X\) is completely decomposable for all pure rank-1 subgroups \(X\) of \(A\), which leads to an order theoretical condition on the types of the summands \(A_ i\). This theorem is settled in the area of finite rank Butler groups, which can be characterized as torsion free images of completely decomposable abelian groups.