Euclidean norm

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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.