Doctoral Dissertations

Date of Award

5-2003

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

David E. Dobbs

Abstract

This work develops concepts related to the going-up property in commutative ring theory. In Chapter 1, we collect some facts in commutative ring theory essential for understanding this work. In Chapter 2, we focus on concepts related to the notion of "quasi-going-up," as defined by Dobbs and Fontana. We define quasi-going-up domains and develop various results for quasi-going-up domains. Such results include a characterization of quasi-going-up domains in terms of going-down domains, results concerning ascent and descent, and various necessary and sufficient conditions for a domain to be a quasi-going-up domain. We next introduce the related concepts of "absolutely quasi-going-up domains" and "universally quasi-going-up domains". We characterize both absolutely quasi-going-up domains and universally quasi-going-up domains. We conclude Chapter 2 by introducing and briefly studying the "quasigoing- up ring" notion, which generalizes the concept of "quasi-going-up domains". In Chapter 3, we define the notion of "generalized going-up" analogously to the notion of "generalized going-down" that was recently introduced by Dobbs, Fontana, and G. Pica vet. We develop analogous results for arbitrary ring homomorphisms satisfying the generalized going-up property. We conclude Chapter 3 with a discussion of 2-chain morphisms ( or subtrusive morphisms) and show that universally 2-chain morphisms descend the generalized going-up property.

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