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.
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.