Prime Ideals in GCD-Domains

Abstract

A GCD-domain is a commutative integral domain in which each pair of elements has a greatest common divisor (g.c.d.). (This is the terminology of Kaplansky [9]. Bourbaki uses the term ''anneau pseudobezoutien" [3, p. 86], while Cohn refers to such rings as "HCF-rings" [4].) The concept of a GCD-domain provides a useful generalization of that of a unique factorization domain (UFD), since several of the standard results for a UFD can be proved in this more general setting (for example, integral closure, some properties of D[X], etc.). Since the class of GCD-domains contains all of the Bezout domains, and in particular, the valuation rings, it is clear that some of the properties of a UFD do not hold in general in a GCD-domain. Among these are complete integral closure, ascending chain condition on principal ideals, and some of the important properties of minimal prime ideals.