Gcd domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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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 $a_{1},a_{2},\ldots ,a_{n}$, 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 $x$.

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 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