# Module over a Lie ring

From Groupprops

## 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 be a Lie ring and be an abelian group. A **module** structure of over is a map:

satisfying the following:

- For every , the map is an endomorphism of the abelian group .
- For every , the map is a homomorphism of groups from the additive group of to .
- For and , we have .

## 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.