Page "Greatest common divisor" Paragraph 86

Note that with this definition, two elements a and b may very well have several greatest common divisors, or none at all.

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

Existence of a gcd is not assured in arbitrary integral domains.

However if R is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains.