Let $R$ be an integral domain and $R^* = R \setdif \{0\}$. Define the equivalence relation on $R \times R^*$:
\[(a, b) \sim (x, y) \enspace\lrimp\enspace ax = by\]for all $(a, b)$, $(x, y) \in R \times R^*$; and write
\[\frac{a}{b} = [(a, b)]\]for all $(a, b) \in R \times R^*$.
The fraction field of $R$ is the field
\[\fract R = (R \times R^*) / {\sim}\]with addition and multiplication defined by
\[\begin{align*} \frac{a}{b} + \frac{x}{y} &= \frac{ay + bx}{by}, \\ \frac{a}{b}\cdot\frac{x}{y} &= \frac{ax}{by} \end{align*}\]for all $a / b$, $x / y \in \fract R.$