# Euclidean norm

## Definition

Let be a commutative unital ring. A Euclidean norm on is a function from the set of nonzero elements of to the set of nonnegative integers, such for that for any with not zero, there exist such that:

and either or . Such a pair is termed a *quotient-remainder pair* for . here is the dividend and is the divisor.

For convenience, we set the norm of zero as .

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 , then the norm of is at least as much as the norm of .
- 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 , .

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