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 given any finite collection of nonzero elements $a_1,a_2,\dots ,a_n$, there exists an element $d$ such that $c|a_i\mathrm{\forall }i$ if and only if $c|d$.

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
• Unique factorization domain