= Derivation of an algebra
{title2=$\operatorname{Der}_k(A)$}
= Derivations of an algebra
{synonym}
= Derivation
{synonym}
For an <associative algebra> $A$ over a <field> $k$, a derivation is a $k$-<linear map> $D:A\to A$ satisfying $D(ab)=D(a)b+aD(b)$. It is the case $M=A$ of a <derivation into a bimodule>; the order of factors matters when $A$ is noncommutative. Substituting $a=b=1$ gives $D(1)=0$. Such maps form a <Lie algebra> under the <commutator> $[D,E]=D\circ E-E\circ D$.
More generally, for a <commutative ring> $R$, a derivation from a commutative $R$-<algebra> $B$ into a $B$-<module> $M$ is an $R$-<linear map> $D:B\to M$ with $D(r)=0$ for $r\in R$ and $D(bb')=bD(b')+b'D(b)$. The second term has this order because the <algebra> is commutative. <Derivations of smooth functions are vector fields> provide a geometric example.
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