# Module over a Lie ring

## Definition

### Symbol-free definition

A module over a Lie ring is an abelian group along with a 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 $[l_1,l_2] \cdot a = l_1 \cdot (l_2 \cdot a) - l_2 \cdot (l_1 \cdot a)$.