# Derivation of a Lie ring

From Groupprops

## Definition

Let be a Lie ring. A **derivation** of is a map satisfying the following two conditions:

- is an endomorphism with respect to the abelian group that is the additive group of .
- satisfies the
*Leibniz rule*for the Lie bracket:

.

When is given the additional structure of a Lie *algebra* over a field or ring , then a *derivation of a Lie algebra* is a derivation that is also a -module map.

The derivations of a Lie ring , themselves form a Lie ring, where the Lie bracket of two derivations is their commutator. This is termed the Lie ring of derivations of and is denoted by . Further, there is a natural homomorphism of Lie rings from to . `Further information: Lie ring acts as derivations by adjoint action`

## Related notions

- Derivation of a non-associative ring generalizes the notion from Lie rings to non-associative rings.