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prime spectrum of a ring: A survey

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Date Issued:
1991
Summary:
This thesis has as its motivation the exploration, on an informal level, of a correspondence between Algebra and Topology. Specifically, it considers the prime spectrum of a ring, that is, the set of prime ideals, endowed with the Zariski topology. Questions posed by M. Atiyah and I. MacDonald in their book, "Introduction to Commutative Algebra", serve as a guideline through most of this work. The final section, however, follows R. Heitmann's paper, "Generating Non-Noetherian Modules Efficiently". This section examines the patch topology on the prime spectrum of a ring where the patch topology has as a closed subbasis the Zariski closed and Zariski quasi-compact open sets. It is proven that the prime spectrum of a ring with the patch topology is a compact Hausdorff space, and several relationships between the patch and Zariski topologies are established. The final section concludes with a technical theorem having a number of interesting corollaries, among which are a stable range theorem and a theorem of Kronecker, both generalized to the non-Noetherian setting.
Title: The prime spectrum of a ring: A survey.
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Name(s): Fernandez, James Stephen
Florida Atlantic University, Degree Grantor
Klingler, Lee, Thesis Advisor
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Issuance: monographic
Date Issued: 1991
Publisher: Florida Atlantic University
Place of Publication: Boca Raton, Fla.
Physical Form: application/pdf
Extent: 60 p.
Language(s): English
Summary: This thesis has as its motivation the exploration, on an informal level, of a correspondence between Algebra and Topology. Specifically, it considers the prime spectrum of a ring, that is, the set of prime ideals, endowed with the Zariski topology. Questions posed by M. Atiyah and I. MacDonald in their book, "Introduction to Commutative Algebra", serve as a guideline through most of this work. The final section, however, follows R. Heitmann's paper, "Generating Non-Noetherian Modules Efficiently". This section examines the patch topology on the prime spectrum of a ring where the patch topology has as a closed subbasis the Zariski closed and Zariski quasi-compact open sets. It is proven that the prime spectrum of a ring with the patch topology is a compact Hausdorff space, and several relationships between the patch and Zariski topologies are established. The final section concludes with a technical theorem having a number of interesting corollaries, among which are a stable range theorem and a theorem of Kronecker, both generalized to the non-Noetherian setting.
Identifier: 14763 (digitool), FADT14763 (IID), fau:11554 (fedora)
Note(s): Thesis (M.S.)--Florida Atlantic University, 1991.
Subject(s): Rings (Algebra)
Held by: Florida Atlantic University Libraries
Persistent Link to This Record: http://purl.flvc.org/fcla/dt/14763
Sublocation: Digital Library
Use and Reproduction: Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Host Institution: FAU
Is Part of Series: Florida Atlantic University Digital Library Collections.