Euclidean domain

An integral domain \(R\) is a Euclidean domain if there exists a function \(f\colon R\setminus\theset{0} \to \ZZ^{\geq 0}\), a Euclidean function, such that for all \(a\in R\) and all nonzero \(b\in R\) there exist \(q, r\in R\) with \(a = bq + r\) and either \(r=0\) or \(f(r) < f(b)\).