Gcd domain: Difference between revisions
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* Given any finite collection of nonzero elements <math>a_1,a_2,\ldots,a_n</math>, there exists an element <math>d</math> such that <math>c|a_i \ \forall \ i</math> if and only if <math>c|d</math> | * Given any finite collection of nonzero elements <math>a_1,a_2,\ldots,a_n</math>, there exists an element <math>d</math> such that <math>c|a_i \ \forall \ i</math> if and only if <math>c|d</math> | ||
* If <math>I</math> is a [[finitely generated ideal]], the intersection of all [[principal ideal]]s containing it is principal. In other words, there exists a ''smallest'' principal ideal containing <math> | * If <math>I</math> is a [[finitely generated ideal]], the intersection of all [[principal ideal]]s containing it is principal. In other words, there exists a ''smallest'' principal ideal containing <math>I</math>. | ||
Note that any two candidates for such an element <math>d</math> must differ multiplicatively by an invertible element, hence we can talk of ''the'' element <math>d</math>. Such an element is termed a | Note that any two candidates for such an element <math>d</math> must differ multiplicatively by an invertible element, hence we can talk of ''the'' element <math>d</math>. Such an element is termed a [[greatest common divisor]] or gcd. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Euclidean domain]] | * [[Weaker than::Euclidean domain]] | ||
* [[Principal ideal domain]] | * [[Weaker than::Principal ideal domain]] | ||
* [[Bezout domain]]: {{proofat|[[Bezout implies gcd]]}} | * [[Weaker than::Bezout domain]]: {{proofat|[[Bezout implies gcd]]}} | ||
* [[Unique factorization domain]]: {{proofat|[[UFD implies gcd]]}} | * [[Weaker than::Unique factorization domain]]: {{proofat|[[UFD implies gcd]]}} | ||
Latest revision as of 01:37, 24 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
Definition with symbols
An integral domain is termed a gcd domain if it satisfies the following equivalent conditions:
- Given any finite collection of nonzero elements , there exists an element such that if and only if
- If is a finitely generated ideal, the intersection of all principal ideals containing it is principal. In other words, there exists a smallest principal ideal containing .
Note that any two candidates for such an element must differ multiplicatively by an invertible element, hence we can talk of the element . Such an element is termed a greatest common divisor or gcd.
Relation with other properties
Stronger properties
- Euclidean domain
- Principal ideal domain
- Bezout domain: For full proof, refer: Bezout implies gcd
- Unique factorization domain: For full proof, refer: UFD implies gcd