Theses and Dissertations

Dissertation

1991

Degree Name

PhD (Doctor of Philosophy)

Mathematics

D. D. Anderson

Abstract

GCD-domains are an important class of integral domains from classical ideal theory. In a GCD-domain, the intersection of any two principal ideals is principal. This property can be generalized in several different ways. A domain for which the intersection of any two invertible ideals is invertible is called a generalized GCD-domain (GGCD-domain). If for elements $a,b \in R - \{0\}$, there is an $n$ = $n(a,b)$ with $a\sp n R\ \cap\ b\sp n R$ principal, we say $R$ is an almost GCD-domain (AGCD-domain). Combining these two definitions, we get an almost generalized GCD-domain (AGGCD-domain)--for $a,b \in R - \{0\}$, there is an $n$ = $n(a,b)$ with $a\sp n R \cap b\sp n R$ invertible.

Anderson and Zafrullah began the study of the first two of these generalizations. They showed that the integral closure of an AGCD-domain is also an AGCD-domain. We show that, in general, an overring of an AGCD-domain need not be an AGCD-domain. Certain special types of overrings do, however, inherit the property. Among these, it is shown, are localizations and LCM-stable overrings. A similar result holds for the AGGCD-domains. Relationships between these classes of domains and the classical domains of ring theory are investigated.

We also investigate how adding the property that $R$ is Noetherian affects an AGCD- or AGGCD-domain. It is shown that a Noetherian AGCD-domain is almost weakly factorial, that is, $R$ = $\cap R\sb P$, where $P$ ranges over all rank one primes of $R$, has finite character and $R$ has torsion t-class group. Similarly, it is shown that a Noetherian AGGCD-domain is weakly Krull, that is, $R$ = $\cap R\sb P$, where $P$ ranges over all rank one primes of $R$, has finite character.

Finally, we consider two additional generalizations defined using the ideal $I\sb n$, where $I\sb n$ = $\{i\sp n\ \vert\ i \in I\}$. A domain $R$ is called a nearly GCD-domain or NGCD-domain (respectively nearly generalized GCD-domain or NGGCD-domain) if for $a,b \in R - \{0\}$, there is an $n$ = $n(a,b)$ with $\lbrack (a,b)\sb n\rbrack \sb t$ principal (respectively invertible).

vi, 74 pages

Bibliography

Includes bibliographical references (pages 73-74).