Euclidean domain: Difference between revisions
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An [[integral domain]] <math>R</math> is termed a '''Euclidean domain''' if there exists a function <math>N</math> from the set of nonzero elements of <math>R</math> to the set of nonnegative integers satisfying the following properties: | An [[integral domain]] <math>R</math> is termed a '''Euclidean domain''' if there exists a function <math>N</math> from the set of nonzero elements of <math>R</math> to the set of nonnegative integers satisfying the following properties: | ||
* <math> | * <math>N(x) = 0</math> if and only if <math>x</math> is a unit | ||
* Given nonzero <math>a</math> and <math>b</math> in <math>R</math>, there exist <math>q</math> and <math>r</math> such that <math>a = qb + r</math> and either <math>r = 0</math> or <math>N(r) < N(b)</math>. | * Given nonzero <math>a</math> and <math>b</math> in <math>R</math>, there exist <math>q</math> and <math>r</math> such that <math>a = qb + r</math> and either <math>r = 0</math> or <math>N(r) < N(b)</math>. | ||
Revision as of 00:17, 17 April 2007
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
Definition with symbols
An integral domain is termed a Euclidean domain if there exists a function from the set of nonzero elements of to the set of nonnegative integers satisfying the following properties:
- if and only if is a unit
- Given nonzero and in , there exist and such that and either or .
We call the dividend, the divisor, the quotient and the remainder.
The definition of Euclidean domain does not require that and be uniquely determined from and . If and a are uniquely determined from and , the integral domain is termed a uniquely Euclidean domain.