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

Statement
Suppose $$L$$ is a Lie ring satisfying the following two conditions:


 * The additive group of $$L$$ is a torsion-free abelian group.
 * The additive group of $$L$$ 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 $$i$$, the upper central series member $$Z^i(L)$$ has a complement in $$L$$ as an additive subgroup of $$L$$. In other words, the additive subgroup $$Z^i(L)$$ is a direct factor of the additive group L.

Facts used

 * 1) uses::Upper central series members are local powering-invariant in Lie ring
 * 2) uses::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 $$L$$ is torsion-free, shows that $$Z^i(L)$$ is a pure subgroup of $$L$$. Fact (2) now shows that it is a direct factor (for the additive structure).