GCD domain

In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor ; i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple.
A GCD domain generalizes a unique [factorization domain] to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals.
GCD domains appear in the following chain of class inclusions:

Properties

Every irreducible element of a GCD domain is prime. A GCD domain is integrally closed, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain.
For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that, or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/''d is an LCM of x'' and y, and vice versa. 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.
If R is a GCD domain, then the polynomial ring R is also a GCD domain.
R is a GCD domain if and only if finite intersections of its principal ideals are principal. In particular,, where is the LCM of and.
For a polynomial in X over a GCD domain, one can define its content as the GCD of all its coefficients. Then the content of a product of polynomials is the product of their contents, as expressed by Gauss's lemma, which is valid over GCD domains.

Examples

A unique factorization domain is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also atomic domains.
A Bézout domain is a GCD domain. Unlike principal ideal domains, a Bézout domain need not be a unique factorization domain; for instance the ring of entire functions is a non-atomic Bézout domain, and there are many other examples. An integral domain is a Prüfer GCD domain if and only if it is a Bézout domain.
If R is a non-atomic GCD domain, then R is an example of a GCD domain that is neither a unique factorization domain nor a Bézout domain ; more generally any ring R has these properties.
A commutative monoid ring is a GCD domain iff is a GCD domain and is a torsion-free cancellative GCD-semigroup. A GCD-semigroup is a semigroup with the additional property that for any and in the semigroup, there exists a such that. In particular, if is an abelian group, then is a GCD domain iff is a GCD domain and is torsion-free.
The ring is not a GCD domain for all square-free integers.

G-GCD domains

Many of the properties of GCD domain carry over to Generalized GCD domains, where principal ideals are generalized to invertible ideals and where the intersection of two invertible ideals is invertible, so that the group of invertible ideals forms a lattice. 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.
Examples of G-GCD domains include GCD domains, polynomial rings over GCD domains, Prüfer domains, and π-domains, which generalizes the GCD property of Bézout domains and unique factorization domains.