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

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Consider the following:

Then, H must be a direct factor.

Related facts

Facts used

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


By Fact (1), G/H is also torsion-free. It is also finitely generated over \mathbb{Z}[\pi^{-1}]. Thus, by Fact (2), it is free as a \mathbb{Z}[\pi^{-1}-module, and we can thus find a complement to H in G.