Derivation

Remarks

Example: let $k$ be a field and $A = k[X]$, and let $D$ be the formal derivative, $D(\sum_i c_i X^i) = \sum_i i c_i X^{i-1}$. It is $k$-linear, and on monomials $D(X^i X^j) = (i + j) X^{i+j-1} = i X^{i-1} X^j + X^i \cdot j X^{j-1}$, which gives clause 2 by linearity. Non-example: on $k[X]$ with $k$ of characteristic $0$, the second derivative $f \mapsto f''$ is linear but fails clause 2, since $(X \cdot X)'' = 2$ while $X'' \cdot X + X \cdot X'' = 0$. Consequences: $D(1) = 0$, hence $D(r \cdot 1) = 0$ for all $r \in R$, and $D(a^n) = n a^{n-1} D(a)$ when $A$ is commutative. If $D_1$ and $D_2$ are derivations, then $D_1 D_2 - D_2 D_1$ is a derivation. More generally, for an $A$-bimodule $M$, a derivation $A \to M$ is an $R$-linear map with $D(ab) = D(a) b + a D(b)$. A derivation of a ring $A$ without a chosen base ring is a derivation over $\Z$, that is, an additive map with the Leibniz rule.