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Integer-valued polynomials and pullbacks of arithmetical rings

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Date Issued:
2006
Summary:
Let D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the n-generator property for D is equivalent to the n-generator property for Int(E, D), which is equivalent to strong (n + 1)-generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (that is, a ring which is locally a chain ring at every maximal ideal). We characterize all Prufer domains R between D[X] and K[X]such that the conductor C of K[X] into R is non-zero. As an application, we show that for n > 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.
Title: Integer-valued polynomials and pullbacks of arithmetical rings.
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Name(s): Boynton, Jason
Florida Atlantic University, Degree Grantor
Klingler, Lee, Thesis Advisor
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Issuance: monographic
Date Issued: 2006
Publisher: Florida Atlantic University
Place of Publication: Boca Raton, Fla.
Physical Form: application/pdf
Extent: 44 p.
Language(s): English
Summary: Let D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the n-generator property for D is equivalent to the n-generator property for Int(E, D), which is equivalent to strong (n + 1)-generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (that is, a ring which is locally a chain ring at every maximal ideal). We characterize all Prufer domains R between D[X] and K[X]such that the conductor C of K[X] into R is non-zero. As an application, we show that for n > 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.
Identifier: 9780542759482 (isbn), 12221 (digitool), FADT12221 (IID), fau:9128 (fedora)
Note(s): Thesis (Ph.D.)--Florida Atlantic University, 2006.
Subject(s): Polynomials
Ideals (Algebra)
Rings of integers
Categories (Mathematics)
Arithmetical algebraic geometry
Held by: Florida Atlantic University Libraries
Persistent Link to This Record: http://purl.flvc.org/fcla/dt/12221
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.
Use and Reproduction: http://rightsstatements.org/vocab/InC/1.0/
Host Institution: FAU
Is Part of Series: Florida Atlantic University Digital Library Collections.