Gcd domain: Difference between revisions

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 $a_{1}, a_{2}, \dots, a_{n}$ , there exists an element $d$ such that $c | a_{i} \forall i$ if and only if $c | d$
If $I$ 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 $I$ .

Note that any two candidates for such an element $d$ must differ multiplicatively by an invertible element, hence we can talk of the element $d$ . 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

@@ Line 8: / Line 8: @@
 * 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>x</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 ''gcd''.
+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==