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).