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.

Kinds of Euclidean norms

Multiplicative Euclidean norm

Further information: 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.