Derivation 2001-12-12 00:59:10 2005-09-15 20:55:33 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \renewcommand{\d}{\mathrm{d}} Let $R$ be a commutative ring. A \emph{derivation} $d$ on an $R$-algebra $A$ into an $A$-module $M$ is an $R$-linear transformation $\d\colon A \to M$ satisfying the properties \begin{itemize} \item $\d(a\x+b\y) = a\,\d\x + b\,\d\y$ \item $\d(\x\cdot \y) = \x \cdot \d\y + \d\x \cdot \y$ \end{itemize} for all $a,b \in R$ and $\x,\y \in A$.