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Polynomials that are integer valued on the image of an integer-valued polynomial
Title
Polynomials that are integer valued on the image of an integer-valued polynomial.
Creator
Marshall, Mario V., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
Let D be an integral domain and f a polynomial that is integer-valued on D. We prove that Int(f(D);D) has the Skolem Property and give a description of its spectrum. For certain discrete valuation domains we give a basis for the ring of integer-valued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integer-valued functions.
Date Issued
2009
PURL
http://purl.flvc.org/FAU/216411
Subject Headings
Polynomials, Ring of integers, Ideals (Algebra)
Format
Document (PDF)
Rings of integer-valued polynomials and derivatives
Title
Rings of integer-valued polynomials and derivatives.
Creator
Villanueva, Yuri., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c...
Show moreFor D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, one-dimensional, analytically irreducible, with finite residue field.
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Date Issued
2012
PURL
http://purl.flvc.org/FAU/3356899
Subject Headings
Rings of integers, Ideals (Algebra), Polynomials, Arithmetic algebraic geometry, Categories (Mathematics), Commutative algebra
Format
Document (PDF)
Pruefer domains, the strong 2-generator property, and integer-valued polynomials
Title
Pruefer domains, the strong 2-generator property, and integer-valued polynomials.
Creator
Roth, Heather., Florida Atlantic University, Klingler, Lee
Abstract/Description
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...
Show moreWe 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).
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Date Issued
2004
PURL
http://purl.flvc.org/fcla/dt/13151
Subject Headings
Prüfer rings, Rings of integers, Polynomials, Ideals (Algebra), Mathematical analysis
Format
Document (PDF)
Integer-valued polynomials and pullbacks of arithmetical rings
Title
Integer-valued polynomials and pullbacks of arithmetical rings.
Creator
Boynton, Jason, Florida Atlantic University, Klingler, Lee
Abstract/Description
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...
Show moreLet 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.
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Date Issued
2006
PURL
http://purl.flvc.org/fcla/dt/12221
Subject Headings
Polynomials, Ideals (Algebra), Rings of integers, Categories (Mathematics), Arithmetical algebraic geometry
Format
Document (PDF)
Maximally Prüfer rings
Title
Maximally Prüfer rings.
Creator
Sharma, Madhav, Klingler, Lee, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong...
Show moreIn this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings.
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Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00004465, http://purl.flvc.org/fau/fd/FA00004465
Subject Headings
Approximation theory, Commutative algebra, Commutative rings, Geometry, Algebraic, Ideals (Algebra), Mathematical analysis, Prüfer rings, Rings (Algebra), Rings of integers
Format
Document (PDF)