Decompositions of torsion-free abelian groups

TL;DR

This paper investigates the structure of torsion-free abelian groups, showing that while finite rank groups have unique maximal decomposable parts, infinite rank groups may lack such parts, but when they exist, they are unique.

Contribution

It extends the understanding of decompositions in torsion-free abelian groups from finite to infinite rank cases, establishing conditions for uniqueness.

Findings

Finite rank torsion-free abelian groups have unique maximal decomposable summands.

Infinite rank groups may lack maximal decomposable summands.

When present, these summands are unique up to isomorphism.

Abstract

It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.