Let be a commutative unital ring. A Euclidean norm on is a function from the set of non-zero divisors of to the set of nonnegative integers, such for that for any with not a zero divisor, 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 divisors as infinity.
Note that when the underlying commutative unital ring is an integral domain, zero divisor can be replaced by zero and the norm function is thus defined on all nonzero elements.
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
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.