# Definition:Euclidean Domain

## Definition

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

Let $\nu: R \setminus \left\{{0_R}\right\} \to \N$ be a function such that

$(1): \quad$ For any $a, b \in R, b \ne 0_R$, there exist $q, r \in R$ with $\nu \left({r}\right) < \nu \left({b}\right)$, or $r = 0_R$ such that:
$a = q \circ b + r$
$(2): \quad$ For any $a, b \in R, b \ne 0_R$:
$\nu \left({a}\right) \le \nu \left({a \circ b}\right)$

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

## Source of Name

This entry was named for Euclid.

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