Let $R$ be a commutative ring. A derivation $d$ on an $R$-algebra $A$ into an $A$-module $M$ is an $R$-linear transformation $\mathrm{d}\colon A\to M$ satisfying the properties

• $\mathrm{d}(a\mathbf{x}+b\mathbf{y})=a\,\mathrm{d}\mathbf{x}+b\,\mathrm{d}\mathbf{y}$

• $\mathrm{d}(\mathbf{x}\cdot\mathbf{y})=\mathbf{x}\cdot\mathrm{d}\mathbf{y}+\mathrm{d}\mathbf{x}\cdot\mathbf{y}$

for all $a,b\in R$ and $\mathbf{x},\mathbf{y}\in A$.