GCD domain

In mathematics, a GCD 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 given two elements. Equivalently, any two elements of R have a least common multiple (LCM).

A GCD domain generalizes a unique factorization domain (UFD) 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 (and in particular if it is Noetherian).

GCD domains appear in the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

gtag('config', 'GA_MEASUREMENT_ID_2', {

 'linker': {
   'domains': ['example-1.com', 'example-2.com']
 }

});

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 (which means that at least one factorization into irreducible elements exists for any nonzero nonunit).
• A Bézout domain (i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike principal ideal domains (where every ideal is principal), 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[X] is an example of a GCD domain that is neither a unique factorization domain (since it is non-atomic) nor a Bézout domain (since X and a non-invertible and non-zero element a of R generate an ideal not containing 1, but 1 is nevertheless a GCD of X and a); more generally any ring R[X₁,...,X_n] has these properties.
• A commutative monoid ring $R[X;S]$ is a GCD domain iff $R$ is a GCD domain and $S$ is a torsion-free cancellative GCD-semigroup. A GCD-semigroup is a semigroup with the additional property that for any $a$ and $b$ in the semigroup $S$, there exists a $c$ such that $(a+S)\cap (b+S)=c+S$. In particular, if $G$ is an abelian group, then $R[X;G]$ is a GCD domain iff $R$ is a GCD domain and $G$ is torsion-free.