# 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

From Groupprops

## Statement

Suppose is a class two Lie ring. Denote by the torsion subgroup of the additive group of . is a fully invariant subgroup of , hence an ideal, hence the quotient is also a class two Lie ring. Suppose that 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 such that is finitely generated as a -module.

Suppose is a characteristic Lie subring of . Then, is a powering-invariant Lie subring of .