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Pruefer domains, the strong 2-generator property, and integer-valued polynomials

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Date Issued:
2004
Summary:
We present several results involving three concepts: Prufer domains, the strong 2-generator property, and integer-valued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2-generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2-generator property . We note that, if D has the strong 2-generator property, then D is a Prufer domain. If Q is the field of fractions of D, and E is a finite nonempty subset of D; we define Int(E, D ) = {f(X) ∈ Q[ X] ∣ f(a) ∈ D for every a ∈ E} to be the ring of integer-valued polynomials on D with respect to the subset E. We show that D is a Prufer domain if and only if Int(E, D) is a Prufer domain. Our main theorem is that Int(E, D) has the strong 2-generator property if and only if D is a Bezout domain (that is, every finitely generated ideal of D is principal).
Title: Pruefer domains, the strong 2-generator property, and integer-valued polynomials.
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Name(s): Roth, Heather.
Florida Atlantic University, Degree grantor
Klingler, Lee, Thesis advisor
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Issuance: monographic
Date Issued: 2004
Publisher: Florida Atlantic University
Place of Publication: Boca Raton, Fla.
Physical Form: application/pdf
Extent: 37 p.
Language(s): English
Summary: We present several results involving three concepts: Prufer domains, the strong 2-generator property, and integer-valued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2-generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2-generator property . We note that, if D has the strong 2-generator property, then D is a Prufer domain. If Q is the field of fractions of D, and E is a finite nonempty subset of D; we define Int(E, D ) = {f(X) ∈ Q[ X] ∣ f(a) ∈ D for every a ∈ E} to be the ring of integer-valued polynomials on D with respect to the subset E. We show that D is a Prufer domain if and only if Int(E, D) is a Prufer domain. Our main theorem is that Int(E, D) has the strong 2-generator property if and only if D is a Bezout domain (that is, every finitely generated ideal of D is principal).
Identifier: 9780496257317 (isbn), 13151 (digitool), FADT13151 (IID), fau:10012 (fedora)
Collection: FAU Electronic Theses and Dissertations Collection
Note(s): Thesis (M.S.)--Florida Atlantic University, 2004.
Charles E. Schmidt College of Science
Subject(s): Prüfer rings
Rings of integers
Polynomials
Ideals (Algebra)
Mathematical analysis
Held by: Florida Atlantic University Libraries
Persistent Link to This Record: http://purl.flvc.org/fcla/dt/13151
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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Owner Institution: FAU
Is Part of Series: Florida Atlantic University Digital Library Collections.