Definition:Euclidean Domain

Definition
Let $(R,+,\circ)$ be an integral domain with zero $0$.

Let $\nu : R \backslash \{0_R\} \to \N$ be a function such that
 * For any $a,b \in R$, $b \neq 0$, there exist $q, r \in R$ with $\nu(r) < \nu(b)$, or $r=0$ such that
 * $ a = q\circ b + r $


 * For any $a,b \in R$, $b \neq 0$,
 * $ \nu(a) \leq \nu (a\circ b) $

Then $\nu$ is called a Euclidean valuation or Euclidean function and $R$ is called a Euclidean ring or Euclidean domain.

Examples

 * The integers are a Euclidean domain with $\nu(x) = |x|$, $x \neq 0$.


 * From Polynomial Forms over Field is Euclidean Domain, the polynomial ring $K[X]$ over a field is Euclidean with valuation $\nu(f) = \deg(f)$, where $\deg(f)$ is the degree of $0 \neq f \in K[X]$

A Euclidean domain is so named because, as an algebraic structure, it sustains the concept of the Euclidean Algorithm.