Euclidean domain

An integral domain \(R\) is Euclidean if there exists a function \(d\colon R\to \ZZ_{\geq 0}\) such that for all \(x,y\in R\) with \(y\neq 0\) there exist \(q,r\in R\) with \(x = qy + r\) and either \(r=0\) or \(d(r) < d(y)\). Such a function \(d\) is a Euclidean function on \(R\).