GCD Domain - Properties

Properties

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.

If R is a GCD domain, then the polynomial ring R is also a GCD domain, and more generally, the group ring R is a GCD domain for any torsion-free commutative group G.

For a polynomial in X over a GCD domain, one can define its contents as the GCD of all its coefficients. Then the contents of a product of polynomials is the product of their contents, as expressed by Gauss's lemma, which is valid over GCD domains.