Definition:Euclidean Domain

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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

For any $a,b \in R$, $b \neq 0_R$, 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$

$ \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.

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 $0 \neq f \in K \left[{X}\right]$

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.