| hroughout, R denotes an integral domain. We say that R is a GCD-domain if each pair of nonzero elements of R has a greatest common divisor.;The purpose of this dissertation is to study algebraic properties of some generalizations of GCD-domains.;Chapter I is an introduction.;In Chapter II, we list several results on GCD-domains, especially those concerning Gauss's Lemma, the prime ideal structure of a GCD-domain in terms of PF-primes, and the PSP-property.;In Chapter III, we restrict our attention to some classes of domains satisfying Gauss's Lemma. We start by recalling definitions of some generalizations. Then we define several new generalizations and establish the relationships among all the classes considered. The central result gives a new characterization of Gauss's Lemma.;In Chapter IV, we study some properties of the class of domains formed by weak GCD-domains. and its subclass formed by MCD-domains. We give a characterization of MCD-domains in terms of principal ideals minimal over v-ideals of finite type. Next we prove that a domain is a GCD-domain if and only if it is a weak GCD-domain satisfying Gauss's Lemma. Then we study overrings of the form ;Finally, in Chapter V, we give new results on... |
