Euclidean norm: Difference between revisions

Latest revision as of 18:25, 23 January 2009

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.

@@ Line 1: / Line 1: @@
 ==Definition==
-Let <math>R</math> be a [[commutative unital ring]]. A Euclidean norm on <math>R</math> is a function <math>N</math> from the set of non-zero divisors of <math>R</math> to the set of nonnegative integers, such for that for any <math>a,b \in R</math> with <math>b</math> not a zero divisor, there exist <math>q,r \in R</math> such that:
+Let <math>R</math> be a [[commutative unital ring]]. A Euclidean norm on <math>R</math> is a function <math>N</math> from the set of nonzero elements of <math>R</math> to the set of nonnegative integers, such for that for any <math>a,b \in R</math> with <math>b</math> not zero, there exist <math>q,r \in R</math> such that:
 <math>a = bq + r</math>
@@ Line 7: / Line 7: @@
 and either <math>r = 0</math> or <math>N(r) < N(b)</math>. Such a pair <math>(q,r)</math> is termed a ''quotient-remainder pair'' for <math>(a,b)</math>. <math>a</math> here is the dividend and <math>b</math> is the divisor.
-For convenience, we set the norm of zero divisors as infinity.
+For convenience, we set the norm of zero as <math>\infty</math>.
-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.
+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==
@@ Line 21: / Line 23: @@
 ===Multiplicative Euclidean norm===
+{{further|[[multiplicative Euclidean norm]]}}
 A Euclidean norm is multiplicative if for any <math>x,y \in R</math>, <math>N(xy) = N(x)N(y)</math>.
 An example of a multiplicative Euclidean norm is the usual absolute value function on the ring of integers.