Free Lie ring on an abelian group

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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.