Gcd domain

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 $$I$$.

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.

Stronger properties

 * Weaker than::Euclidean domain
 * Weaker than::Principal ideal domain
 * Weaker than::Bezout domain:
 * Weaker than::Unique factorization domain: