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

From Groupprops

## Definition

A **Lie ring** 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 is an 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 of the set of prime numbers such that the additive group of is a finitely generated as a module over the ring . Another way of putting it is that there is a finite subset of such that the -powered subgroup generated by is the whole group .