Derivation of a non-associative ring

Definition
Let $$R$$ be a non-associative ring (i.e., a not necessarily associative ring). A function $$d:R \to R$$ is termed a derivation of $$R$$ if it satisfies the following two conditions:


 * 1) $$d$$ is an endomorphism of the additive group of $$R$$.
 * 2) $$d$$ satisfies the Leibniz rule for multiplication: If $$*$$ denotes the multiplication, then:

$$\! d(x * y) = d(x) * y + x * d(y) \ \forall x,y \in R$$.

The derivations of any ring form a Lie ring with the Lie bracket given by:

$$[d_1,d_2] := d_1 \circ d_2 - d_2 \circ d_1$$

Two special cases of interest are derivation of a Lie ring (where the elements themselves act as derivations by the adjoint action) and derivation of an associative ring (where the elements themselves act as derivations by the commutator).