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 Title
 ANNIHILATORS AND A + B RINGS.
 Creator
 Epstein, Alexandra Nicole, Klingler, Lee, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

A + B rings are constructed from a ring A and nonempty set of prime ideals of A. Initially, these rings were created to provide examples of reduced rings which satisfy certain annihilator conditions. We describe precisely when A + B rings have these properties, based on the ring A and set of prime ideals of A. We continue by giving necessary and su cient conditions for A + B rings to have various other properties. We also consider annihilators in the context of frames of ideals of reduced rings.
 Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013588
 Subject Headings
 Rings (Algebra)
 Format
 Document (PDF)
 Title
 The prime spectrum of a ring: A survey.
 Creator
 Fernandez, James Stephen, Florida Atlantic University, Klingler, Lee
 Abstract/Description

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 NonNoetherian Modules...
Show moreThis 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 NonNoetherian 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 quasicompact 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 nonNoetherian setting.
Show less  Date Issued
 1991
 PURL
 http://purl.flvc.org/fcla/dt/14763
 Subject Headings
 Rings (Algebra)
 Format
 Document (PDF)
 Title
 Unique decomposition of direct sums of ideals.
 Creator
 Ay, Basak., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any Rmodule which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 13 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of one...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any Rmodule which decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains. In Chapters 13 the UDI property for reduced Noetherian rings is characterized. In Chapter 4 it is shown that overrings of onedimensional reduced commutative Noetherian rings with the UDI property have the UDI property, also. In Chapter 5 we show that the UDI property implies the KrullSchmidt property for direct sums of torsionfree rank one modules for a reduced local commutative Noetherian onedimensional ring R.
Show less  Date Issued
 2010
 PURL
 http://purl.flvc.org/FAU/2683133
 Subject Headings
 Algebraic number theory, Modules (Algebra), Noetherian rings, Commutative rings, Algebra, Abstract
 Format
 Document (PDF)
 Title
 Minimal zerodimensional extensions.
 Creator
 Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The structure of minimal zerodimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a onetoone correspondence between isomorphism classes of minimal zerodimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zerodimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a onetoone correspondence between isomorphism classes of minimal zerodimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zerodimensional extensions of general ZPIrings.
Show less  Date Issued
 2009
 PURL
 http://purl.flvc.org/FAU/210447
 Subject Headings
 Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
 Format
 Document (PDF)
 Title
 Pruefer domains, the strong 2generator property, and integervalued polynomials.
 Creator
 Roth, Heather., Florida Atlantic University, Klingler, Lee
 Abstract/Description

We present several results involving three concepts: Prufer domains, the strong 2generator property, and integervalued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2generated 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 2generator property . We note that, if D has the strong 2generator property, then D is a...
Show moreWe present several results involving three concepts: Prufer domains, the strong 2generator property, and integervalued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2generated 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 2generator property . We note that, if D has the strong 2generator 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 integervalued 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 2generator property if and only if D is a Bezout domain (that is, every finitely generated ideal of D is principal).
Show less  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)
 Title
 Polynomials that are integer valued on the image of an integervalued 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 integervalued 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 integervalued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integervalued functions.
 Date Issued
 2009
 PURL
 http://purl.flvc.org/FAU/216411
 Subject Headings
 Polynomials, Ring of integers, Ideals (Algebra)
 Format
 Document (PDF)
 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 Pruferlike 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 Pruferlike 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.
Show less  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)
 Title
 AuslanderReiten theory for systems of submodule embeddings.
 Creator
 Moore, Audrey., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In this dissertation, we will investigate aspects of AuslanderReiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute AuslanderReiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and RingelTachikawa Theorem which states that for an artinian ring R of finite...
Show moreIn this dissertation, we will investigate aspects of AuslanderReiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute AuslanderReiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and RingelTachikawa Theorem which states that for an artinian ring R of finite representation type, each Rmodule is a direct sum of finitelength indecomposable Rmodules. In cases where this applies, the indecomposable objects obtained in the AuslanderReiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise CalabiYau dimension of such a category.
Show less  Date Issued
 2009
 PURL
 http://purl.flvc.org/fcla/dt/210496
 Subject Headings
 Artin algebras, Rings (Algebra), Representation of algebras, Embeddings (Mathematics), Linear algebraic groups
 Format
 Document (PDF)
 Title
 Rings of integervalued 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 integervalued 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 integervalued 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, onedimensional, analytically irreducible, with finite residue field.
Show less  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)
 Title
 A study of divisors and algebras on a double cover of the affine plane.
 Creator
 Bulj, Djordje., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x  1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed.
 Date Issued
 2012
 PURL
 http://purl.flvc.org/FAU/3355618
 Subject Headings
 Algebraic number theory, Geometry, Data processing, Noncommutative differential geometry, Mathematical physics, Curves, Algebraic, Commutative rings
 Format
 Document (PDF)
 Title
 Integervalued 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 ngenerator property for D is equivalent to the ngenerator 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 ngenerator property for D is equivalent to the ngenerator 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 nonzero. As an application, we show that for n > 2, such a ring R has the ngenerator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.
Show less  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)