# Completely decomposable direct summands of torsion--free abelian groups of finite rank

## TL;DR

This paper proves that finite rank torsion-free abelian groups can be uniquely decomposed into a completely decomposable part and a part with no rank 1 summand, with the decompositions being unique up to isomorphism or near-isomorphism.

## Contribution

It establishes the existence and uniqueness of a specific decomposition for finite rank torsion-free abelian groups, clarifying their structural properties.

## Key findings

- Existence of a decomposition into a completely decomposable and a non-rank-1 summand
- Uniqueness of the decomposable part up to isomorphism
- Uniqueness of the remaining part up to near-isomorphism

## Abstract

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.