Definition:Euclidean Domain

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

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


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

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 $\nu \left({x}\right) = \left|{x}\right|$, $x \neq 0$.


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

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