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

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Statement

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.