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
Jump to: navigation, search

Statement

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

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 <amth>L</math>.

Facts used

  1. Upper central series members are local powering-invariant in Lie ring
  2. 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).