Definition:Euclidean Domain

Definition
Let $\struct {R, +, \circ}$ be an integral domain with zero $0_R$.

Let there exist a mapping $\nu: R \setminus \set {0_R} \to \N$ with the properties:
 * $(1): \quad$ For all $a, b \in R, b \ne 0_R$, there exist $q, r \in R$ with $\map \nu r < \map \nu b$, or $r = 0_R$ such that:
 * $a = q \circ b + r$
 * $(2): \quad$ For all $a, b \in R, b \ne 0_R$:
 * $\map \nu a \le \map \nu {a \circ b}$

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