GCD domain

In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor (GCD). Equivalently, any two non-zero elements of R have a least common multiple (LCM).

A GCD domain generalizes a unique factorization domain 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:

Every irreducible element of a GCD domain is prime (however irreducible elements need not exist, even if the GCD domain is not a field). 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 a LCM m of x and y can be chosen such that dm = xy, or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is a 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.

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