# Characteristic implies powering-invariant in class two Lie ring whose torsion-free part is finitely generated as a module over the ring of integers localized at a set of primes

Suppose $L$ is a class two Lie ring. Denote by $T$ the torsion subgroup of the additive group of $L$. $T$ is a fully invariant subgroup of $L$, hence an ideal, hence the quotient $L/T$ is also a class two Lie ring. Suppose that $L/T$ is a Lie ring whose additive group is finitely generated as a module over the ring of integers localized at a set of primes. Explicitly, there exists a prime set $\pi$ such that $L/T$ is finitely generated as a $\mathbb{Z}[\pi^{-1}]$-module.
Suppose $M$ is a characteristic Lie subring of $L$. Then, $M$ is a powering-invariant Lie subring of $L$.