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.