# Pure subgroup implies direct factor in torsion-free abelian group that is finitely generated as a module over the ring of integers localized at a set of primes

From Groupprops

## Contents

## Statement

Consider the following:

- A group that is a torsion-free abelian group that is also an abelian group that is finitely generated as a module over the ring of integers localized at a set of primes. Explicitly, this means that is torsion-free and is finitely generated as a module over where is a set of prime numbers. By the structure theorem for finitely generated modules over principal ideal domains, is a finitely generated
*free*module over . - is a pure subgroup of .

Then, must be a direct factor.

## Related facts

## Facts used

- Equivalence of definitions of pure subgroup of torsion-free abelian group
- Structure theorem for finitely generated modules over principal ideal domains

## Proof

By Fact (1), is also torsion-free. It is also finitely generated over . Thus, by Fact (2), it is free as a -module, and we can thus find a complement to in .