Let \(R\) be an integral domain. On \(R \times (R\setminus\theset{0})\) define \((a,s)\sim (b, t)\) if \(at = bs\), and write \(a/s\) for the class of \((a,s)\). The field of fractions of \(R\) is the set of classes \[ \ff(R) \coloneqq \bigl(R \times (R\setminus\theset{0})\bigr)/{\sim}, \] with operations \(\frac as+\frac bt=\frac{at+bs}{st}\) and \(\frac as\cdot\frac bt=\frac{ab}{st}\).
Transitivity of \(\sim\) uses that \(R\) has no nonzero zero divisors: if \(at=bs\) and \(bu=ct\), then \(aut=bsu=cst\), so \(t(au-cs)=0\), and \(t\neq0\) gives \(au=cs\). The same hypothesis gives \(st\neq 0\), so the operations are defined on \(R\times(R\setminus\theset{0})\).