# Free Lie ring on an abelian group

## Definition

Let $A$ be an abelian group. The free Lie ring on $A$ is defined as a Lie ring $\mathcal{L}(A)$ along with an embedding of $A$ as a subgroup of the additive group of $\mathcal{L}(A)$ such that, for any Lie ring $M$ and any group homomorphism $\varphi$ from $A$ to $M$ $\varphi$ extends uniquely to a homomorphism of Lie rings from $\mathcal{L}(A)$ to $M$.

Note that this notion is quite different from that of the associated Lie ring of a group. The associated Lie ring, when applied to an abelian group $A$, would give an abelian Lie ring whose additive group is precisely $A$. The free Lie ring, on the other hand, would have separate elements for various brackets of things in $A$, set to be equal only based on the rules of addition in $A$ and the identities that Lie rings must satisfy.