Indecomposable Decompositions of Torsion-free Abelian Groups

TL;DR

This paper investigates the possible partitions of a torsion-free abelian group's rank that can result from its indecomposable decompositions, aiming to characterize which sets of partitions are realizable.

Contribution

It provides a characterization of the sets of partitions of the group's rank that can occur from indecomposable decompositions of torsion-free abelian groups.

Findings

Identifies conditions under which certain partitions can arise

Characterizes all possible sets of partitions for given ranks

Provides a framework for understanding indecomposable decompositions

Abstract

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.