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 two given 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:
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"["X"1...,"X""n"] is also a GCD domain.
R is a GCD domain if and only if finite intersections of its principal ideals are principal. In particular, formula_1, where formula_2 is the LCM of formula_3 and formula_4.
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.