# Upper central series members are additively complemented in torsion-free Lie ring whose additive group is finitely generated as a module over the ring of integers localized at a set of primes

From Groupprops

## Statement

Suppose is a Lie ring satisfying the following two conditions:

- The additive group of is a torsion-free abelian group.
- The additive group of is an abelian group that is finitely generated as a module over the ring of integers localized at a set of primes.

Then, for any positive integer , the upper central series member has a complement in as an *additive* subgroup of . In other words, the additive subgroup is a direct factor of the additive group <amth>L</math>.

## Facts used

- Upper central series members are local powering-invariant in Lie ring
- 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

## Proof

Fact (1), along with the condition that is torsion-free, shows that is a pure subgroup of . Fact (2) now shows that it is a direct factor (for the additive structure).