Euclidean norm

Definition
Let $$R$$ be a commutative unital ring. A Euclidean norm on $$R$$ is a function $$N$$ from the set of nonzero elements of $$R$$ to the set of nonnegative integers, such for that for any $$a,b \in R$$ with $$b$$ not zero, there exist $$q,r \in R$$ such that:

$$a = bq + r$$

and either $$r = 0$$ or $$N(r) < N(b)$$. Such a pair $$(q,r)$$ is termed a quotient-remainder pair for $$(a,b)$$. $$a$$ here is the dividend and $$b$$ is the divisor.

For convenience, we set the norm of zero as $$\infty$$.

Note that we often assume the underlying commutative unital ring to be an integral domain.

A ring which admits a Euclidean norm is termed a Euclidean ring, and an integral domain which admits a Euclidean norm is termed a Euclidean domain.

Facts
The following can be readily verified for a Euclidean norm:


 * If $$a|b$$, then the norm of $$b$$ is at least as much as the norm of $$a$$.
 * The units have the lowest possible Euclidean norm.

Multiplicative Euclidean norm
A Euclidean norm is multiplicative if for any $$x,y \in R$$, $$N(xy) = N(x)N(y)$$.

An example of a multiplicative Euclidean norm is the usual absolute value function on the ring of integers.