Module over a Lie ring

Symbol-free definition
A module over a Lie ring is an abelian group along with a defining ingredient::homomorphism of Lie rings from the Lie ring to the endomorphism ring of the abelian group, with the Lie bracket given by the additive commutator.

Definition with symbols
Let $$L$$ be a Lie ring and $$A$$ be an abelian group. A module structure of $$A$$ over $$L$$ is a map:

$$\cdot:L \times A \to A$$

satisfying the following:


 * 1) For every $$l \in L$$, the map $$a \mapsto l \cdot a$$ is an endomorphism of the abelian group $$A$$.
 * 2) For every $$a \in A$$, the map $$l \mapsto l \cdot a$$ is a homomorphism of groups from the additive group of $$L$$ to $$A$$.
 * 3) For $$l_1,l_2 \in L$$ and $$a \in A$$, we have $$[l_1,l_2] \cdot a = l_1 \cdot (l_2 \cdot a) - l_2 \cdot (l_1 \cdot a)$$.

Facts

 * The notion of module over a Lie ring is closely related to the notion of group action. While groups act on sets, Lie rings act on abelian groups.
 * Just as we can have a group acting on another group by automorphisms, we can have a module over a Lie ring that is itself a Lie ring, where every element of the acting Lie ring acts as derivations. This is equivalent to a homomorphism from the acting Lie ring to the Lie ring of derivations of the Lie ring being acted upon.
 * Just as a group acts as automorphisms by conjugation, i.e., via inner automorphisms, we have that a Lie ring acts as derivations by adjoint action.