Let $R$ be an integral domain and $R^*=R\setminus\{0\}$. 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 $a/b$, $x/y\in\fract R.$