# Definitions > [!NOTE] Definition (Euclidean Domain) > Let $R$ be an [[Integral Domain|integral domain]]. We say that a [[Function|function]] $\partial:R\setminus\{ 0 \}\to \mathbb{N}$ is a Euclidean function if > > (EF1) for all $a,b\in R \setminus \{ 0 \}$, if $a$ [[Divisibility|divides]] $b$ then $\partial(a)\leq \partial(b)$ (or equivalently, for all $c\in R\setminus \{ 0 \}$, $f(a)\leq f(ac)$); > > (EF2) and for all $a,b\in R$ with $b \neq 0$, there exists $q,r\in R$ such that $a=qb+r$ and either $r=0$ or $\partial(r)<\partial(b)$. > > A Euclidean domain (or Euclidean ring) is a pair $(R,\partial)$ where $R$ is an integral domain and $\partial$ a Euclidean function. **Remark**: (EF2) alone suffices to define a Euclidean domain, since any Euclidean function satisfying (EF2) can be normalised [^1] to satisfy (EF1) by defining $g(a)=\min_{x\in R \setminus \{ 0 \}} f(ax)$. # Properties # Reference(s) [^1]: Rogers, K. (1971). The Axioms for Euclidean Domains. _The American Mathematical Monthly_, _78_(10), 1127–1128. https://doi.org/10.1080/00029890.1971.11992960