In mathematics, a GCD domain (sometimes called just domain) is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements.
A GCD domain is integrally closed, and every nonzero element is primal.
It follows that the operations of GCD and LCM make the quotient R/~ into a distributive lattice, where "~" denotes the equivalence relation of being associate elements.
The equivalence between the existence of GCDs and the existence of LCMs is not a corollary of the similar result on complete lattices, as the quotient R/~ need not be a complete lattice for a GCD domain R.[citation needed] If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain.
In GCD rings, ideals are invertible if and only if they are principal, meaning the GCD and LCM operations can also be treated as operations on invertible ideals.