Euclidean domain: Difference between revisions

Revision as of 16:53, 22 January 2009

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

An integral domain is said to be Euclidean if it admits a Euclidean norm.

Definition with symbols

An integral domain $R$ is termed a Euclidean domain if there exists a function $N$ from the set of nonzero elements of $R$ to the set of nonnegative integers satisfying the following properties:

$N(x)=0$ if and only if $x$ is a unit
Given nonzero $a$ and $b$ in $R$ , there exist $q$ and $r$ such that $a=qb+r$ and either $r=0$ or $N(r)<N(b)$ .

We call $a$ the dividend, $b$ the divisor, $q$ the quotient and $r$ the remainder.

Such a function $N$ is called a Euclidean norm on $R$ .

The definition of Euclidean domain does not require that $q$ and $r$ be uniquely determined from $a$ and $b$ . If $q$ and $r$ are uniquely determined from $a$ and $b$ , the integral domain is termed a uniquely Euclidean domain.

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Polynomial-closedness

This property of commutative unital rings is not closed under passing to the polynomial ring

The polynomial ring over a Euclidean domain need not be a Euclidean domain. One example is the polynomial ring with integer coefficients, which is not a Euclidean domain; another example is the polynomial ring in two variables over a field (which can be viewed as the polynomial ring in one variable, over the polynomial ring over a field).

@@ Line 15: / Line 15: @@
 We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.
+Such a function <math>N</math> is called a ''Euclidean norm'' on <math>R</math>.
 The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].