Indecomposable Decompositions of Torsion-free Abelian Groups

An indecomposable decomposition of a torsion-free abelian group \(G\) of rank \(n\) is a decomposition \(G=A_1\oplus\cdots\oplus A_t\) where \(A_i\) is indecomposable of rank \(r_i\) so that \(\sum_i r_i=n\) is a partition of \(n\). The group \(G\) may have decompositions that result in different partitions of \(n\). We address the problem of characterising those sets of partitions of \(n\) which can arise from indecomposable decompositions of a torsion-free abelian group.