Euclidean domain (reviewed)

An Euclidean domain is an integral domain $R$ with a function \[ \nu: R-\{0\}\to \Z_{\geq 0} \] such that for all $a\in R$ and all $b\in R-\{0\}$, there exists $q,r\in R$ such that $a=bq+r$ and $r=0$ or $\nu(r)<\nu(b)$.

Euclidean domains are principal ideal domains, and hence, unique factorization domains.