Derivation of an associative algebra

A derivation of an associative algebra $A$ over a field $k$ is a $k$ -linear map $\delta:A\to A$ satisfying the Leibniz rule

\delta(ab)=\delta(a)b+a\delta(b)\qquad(a,b\in A).

For a unital algebra, this implies $\delta(1)=0$ .

Examples

Ordinary differentiation on polynomial functions is a derivation. For fixed $c\in A$ , the commutator $a\mapsto ca-ac$ is a derivation, since its two middle terms cancel in the Leibniz identity. Derivations of dense subalgebras of operator algebras need not be bounded or extend to the whole completion.