Lie ring whose additive group is finitely generated as a module over the ring of integers localized at a set of primes

Definition
A Lie ring $$L$$ is termed a Lie ring whose additive group is finitely generated as a module over the ring of integers localized at a set of primes if the additive group of $$L$$ is an defining ingredient::abelian group that is finitely generated as a module over the ring of integers localized at a set of primes. Explicitly: there is a (possibly empty, possibly finite, possibly infinite) subset $$\pi$$ of the set of prime numbers such that the additive group of $$L$$ is a finitely generated as a module over the ring $$\mathbb{Z}[\pi^{-1}]$$. Another way of putting it is that there is a finite subset $$S$$ of $$L$$ such that the $$\pi$$-powered subgroup generated by $$S$$ is the whole group $$L$$.