Derivation of a Lie ring

Definition
Let $$L$$ be a Lie ring. A derivation of $$L$$ is a map $$d:L \to L$$ satisfying the following two conditions:


 * $$d$$ is an defining ingredient::endomorphism with respect to the abelian group that is the additive group of $$L$$.
 * $$d$$ satisfies the Leibniz rule for the Lie bracket:

$$d[x,y] = [dx,y] + [x,dy]$$.

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

The derivations of a Lie ring $$L$$, themselves form a Lie ring, where the Lie bracket of two derivations is their commutator. This is termed the Lie ring of derivations of $$L$$ and is denoted by $$\operatorname{Der}(L)$$. Further, there is a natural homomorphism of Lie rings from $$L$$ to $$\operatorname{Der}(L)$$.

Related notions

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