Doctor of Philosophy, The Ohio State University, 2017, Mathematics

Skolem properties describe how well ideals of rings of integer-valued polynomials are characterized by their images under evaluation maps. They are usually defined only for finitely generated ideals. Evaluation is sensible for any ring made of polynomials, and it usually makes sense in the context of rational functions. We generalize the notion of a Skolem property to these broader settings. We give several examples of rings exhibiting these properties, and we extend many of the results about Skolem properties of rings of integer-valued polynomials to rings comprising polynomials. We examine rings in which the interesting values occur at only a finite collection of points. We demonstrate that such rings have the almost strong Skolem property, and we use that result to characterize when they are Prufer domains. We also consider n-generator properties in that setting. Making signicant progress toward classifying the Skolem properties for all integrally closed rings of polynomials, we consider valuation domains on K(x), for some field K, contracted to K[x]. In this setting we characterize the almost Skolem property. We also extend the notion of a Skolem closure so that it is a semistar operation, and we demonstrate that it is more natural to consider the Skolem property as a property of star ideals rather than one of finitely generated ideals. We end with an application of this new perspective to the classical ring of integer-valued polynomials Int(Z), answering the open question: What is the largest class of ideals on which Int(Z) has the (strong) Skolem property?

Committee: K. Alan Loper (Advisor); Ivo Herzog (Committee Member); Cosmin Roman (Committee Member) Subjects: Mathematics

Doctor of Philosophy (PhD), Ohio University, 2005, Mathematics (Arts and Sciences)

A right R-module M is called CS if every submodule of M is essential in a direct summand of M. In this dissertation, we study certain classes of CS and Σ-CS rings and modules. A ring R is called right (left) max-min CS if every maximal closed right (left) ideal with nonzero left (right) annihilator and every minimal closed right (left) ideal of R is a direct summand of R. Among other results, it is shown that if R is a nondomain prime ring, then R is right nonsingular, right max-min CS with a uniform right ideal if and only if R is a left nonsingular, left max-min CS with a uniform left ideal. This result gives, in particular, Huynh, Jain and Lopez-Permouth Theorem for prime rings of finite uniform dimension. Also we show that a nondomain right nonsingular prime ring with a uniform right ideal is right finitely Σ-min-CS if every finitely generated right ideal of R is min CS. Jain, Kanwar and Lopez-Permouth characterized right nonsingular semiperfect right CS rings. We obtain the structure of right nonsingular semiperfect right min CS rings with a uniform right ideal. It is shown that such rings are direct sums of indecomposable right CS rings and a ring with no uniform right ideal. As a consequence, we show that an indecomposable right nonsingular semiperfect ring is right CS if and only if it is min CS with a uniform right ideal. We generalize this result to endomorphism rings of nonsingular semiperfect progenerator min CS modules with a uniform submodule. It is known that every Σ-CS module is a direct sum of uniform modules and countably Σ-CS modules need not be Σ-CS. A sufficient condition that guarantees a countably Σ-CS module, which is a direct sum of uniform modules, to be Σ-CS has been obtained.

Committee: S. Jain (Advisor) Subjects: Mathematics; Mathematics

Master of Science, The Ohio State University, 2024, Environment and Natural Resources

Ring-necked Pheasants (Phasianus colchicus, herein referred to as pheasant) are an introduced game bird that occupy a contemporary niche in Ohio's agricultural ecosystems, serving as an analogue for the native prairie Galliformes. After their introduction in the 1800's pheasants reached a peak density in the 1930's and then began a steady decline. This decline is attributable to the advent of commercial farming and crop subsidies introduced by Roosevelt's New Deal (1933). These subsidies led to drastic land use change in rural Ohio, replacing grassland and fallow fields with large commercial farms. Like the rest of North America, loss of nesting and winter habitat to commercial agriculture has led to the decline of pheasants in Ohio. This loss of habitat has also led to fragmentation and reduced habitat connectivity between suitable patches. Our current understanding of pheasants' response to land cover lacks the context of scale and habitat connectivity. These concepts are important for conservation as changes in arrangement and surrounding cover type may render some habitat unusable despite being the preferred cover type. My objectives were to find suitable patches, connections between them, and areas that would improve connectivity. To better inform the conservation of pheasant, I investigated pheasants' response to cover type and its scale of effect along with habitat connectivity. Using a novel multiscale framework, I analyzed landscape suitability by modeling the influence of cover types on pheasant occupancy and density. To quantify habitat connectivity, I used combination of circuit theory and graph theory to find areas of high importance for connectivity. Conservation Reserve Programs and grassland were both positively related to pheasant occupancy and density at a relatively fine scale, and developed areas and forests had a large negative impact at a broad scale. Additionally, the majority of the state has a low degree of habitat connectivity for pheasant. (open full item for complete abstract)

Committee: William Peterman (Advisor); Stephen Matthews (Committee Member); Robert Gates (Advisor) Subjects: Ecology; Wildlife Conservation; Wildlife Management

Master of Arts (MA), Bowling Green State University, 2023, Mathematics

If a group Γ acts on a ring R then the ring of invariants RΓ is the set of all elements in R that are not changed by the action of Γ. In this paper we recall a few general results from invariant theory and give explicit examples of computations that can be done. More precisely, we compute the ring of invariants and the Hilbert series for the action of cyclic group Cn and the dihedral group Dn on C[X1, X2]. We also investigate the action of S4 on C[Xij | 1 ≤ i < j ≤ 4].

Committee: Mihai Staic Ph.D. (Committee Chair); Benjamin Ward Ph.D. (Committee Member) Subjects: Mathematics

PHD, Kent State University, 2021, College of Arts and Sciences / Department of Mathematical Sciences

Linear preserver problems concern the characterization of linear operators on matrix spaces that leave invariant certain functions, subsets, relations, etc. We present several linear preserver problems whose solutions may be considered nonstandard since they differ significantly from classical results. In addition, we also discuss several related linear preserver problems with standard solutions to highlight the phenomena observed.

Committee: Mikhail Chebotar (Advisor); Joanne Caniglia (Committee Member); Feodor Dragan (Committee Member); Mark L. Lewis (Committee Member); Dmitry Ryabogin (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2004, Mathematics

We introduce the notions of the Baer and the quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property. Every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the extending property and the FI-extending property are exhibited and it is shown that a module M is (quasi-) Baer and (FI-) 퓚-cononsingular if and only if it is (FI-) extending and (FI-) 퓚-nonsingular. While we show that direct sums of (quasi-) Baer modules are not (quasi-) Baer, we prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and that every free (projective) module over a quasi-Baer ring is always a quasi-Baer module. Some results, related to direct sums of Baer modules and direct sums of quasi-Baer modules, are also included. A ring over which every module is Baer is shown to be precisely a semisimple Artinian ring. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. A characterization for this to hold in the Baer modules case is obtained. We provide a type theory of Baer modules and decomposition of a Baer module into into five types, similar to the one provided by Kaplansky for the Baer rings case. This type theory and type decomposition is applied, in particular, to all nonsingular extending modules. Applications of the results obtained are included.

Committee: Syed Rizvi (Advisor) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2023, Mathematics

Persistent homology has been an important tool in topological and geometrical data analysis to study the shape of data. However, its ability to differentiate between various datasets is limited. To expand and enhance our toolkit, we study persistent invariants that arise from homotopy groups, rational homotopy groups, the cohomology ring, the Lyusternik-Schnirelmann (LS) category, and chain complexes, which can be more discriminative than persistent homology. Chapter 2 provides the necessary background from metric geometry, algebraic topology, and persistent theory. Chapter 3 discusses persistent homotopy groups of compact metric spaces, with a focus on the persistent fundamental groups, for which we obtain a more precise description via a persistent version of the notion of discrete fundamental groups due to Berestovskii-Plaut. Under fairly mild assumptions on the spaces, we prove that the persistent fundamental group admits a tree structure that encodes more information than its persistent homology counterpart. The rationalization of the persistent homotopy groups is also considered and completely characterized for the circle by invoking the results of Adamaszek-Adams and Serre. We establish that persistent homotopy groups enjoy stability in the Gromov-Hausdorff sense. Chapter 4 examines several persistent invariants that capture ring-theoretic information about the evolution of the cohomology structure across a filtration. The first one is the persistent cup-length invariant, which is a persistent version of the standard cup-length invariant and is computable from representative cocycles in polynomial time. The second one is the persistent LS-category invariant. Although not directly defined using the cup product operation, the persistent LS-category invariant is closely related to the persistent cup-length invariant by having the latter as a pointwise lower bound. The third one is the persistent cup module, which absorbs the cup prod (open full item for complete abstract)

Committee: Facundo Mémoli (Advisor); Matthew Kahle (Committee Member); Jean-Francois Lafont (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2023, Mathematics

As objects that appear throughout mathematics, integer-valued polynomials have been studied extensively. However, integer-valued rational functions are a much less studied generalization. We consider the set of integer-valued rational functions over an integral domain as a ring and study the ring-theoretic properties of such rings. We explore when rings of integer-valued rational functions are Bezout domains, Prufer domains, and globalized pseudovaluation domains. We completely classify when the ring of integer-valued rational functions over a valuation domain is a Prufer domain and when it is a Bezout domain. We extend the classification of when rings of integer-valued rational functions are Prufer domains. This includes a family of rings of integer-valued rational functions that are Prufer domains, as well as a family of integer-valued rational functions that are not Prufer domains. We determine that the classification of when rings of integer-valued rational functions are Prufer domains is not analogous to the interpolation domain classification of when rings of integer-valued polynomials are Prufer domains. We also show some conditions under which the ring of integer-valued rational functions is a globalized pseudovaluation domain. We also prove that even if a pseudovaluation domain has an associated valuation domain over which the ring of integer-valued rational functions is a Prufer domain, the ring of integer-valued rational functions over the pseudovaluation domain is not guaranteed to be a globalized pseudovaluation domain. Because rings of integer-valued rational functions are rings of functions, we can study their properties with respect to evaluation. These properties include the Skolem property and its generalizations, which are properties concerning when ideals are able to be distinguished using evaluation. We connect the Skolem property to the maximal spectrum of a ring of integer-valued rational functions. This is then generalized using st (open full item for complete abstract)

Committee: K. Alan Loper (Advisor); Cosmin Roman (Committee Member); Ivo Herzog (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2021, Mathematics

The aim of this dissertation is three-fold: (i) we construct a natural highly homotopy coherent operad structure on the derivatives of the identity functor on structured ring spectra which can be described as algebras over an operad O in spectra, (ii) we prove that every connected O-algebra has a naturally occurring left action of the derivatives of the identity, and (iii) we show that there is a naturally occurring weak equivalence of highly homotopy coherent operads between the derivatives of the identity on O-algebras and the operad O. Along the way, we introduce the notion of N-colored operads with levels which, by construction, provides a precise algebraic framework for working with and comparing highly homotopy coherent operads, operads, and their algebras. We also show that similar techniques may be used to provide a new description of an operad structure for the Goodwillie derivatives of the identity in spaces and describe an explicit comparison map from spaces to algebras over such operad.

Committee: John Harper (Advisor); Niles Johnson (Committee Member); Crichton Ogle (Committee Member) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2013, Mathematics

A ring is said to be clean if each element in the ring can be written as the sum of a unit and an idempotent of the ring. More generally, an element in a ring is said to be clean if it can be written as the sum of a unit and an idempotent of the ring. The notion of a clean ring was introduced by Nicholson in his 1977 study of lifting idempotents and exchange rings, and these rings have since been studied by many different authors. In our study of clean rings, we classify the rings that consist entirely of units, idempotents, and quasiregular elements. It is well known that the units, idempotents, and quasiregular elements of any ring are clean. Therefore any ring that consists entirely of these types of elements is clean. We prove that a ring consists entirely of units, idempotents, and quasiregular elements if and only if it is a boolean ring, a local ring, isomorphic to the direct product of two division rings, isomorphic to the full matrix ring M2(D) for some division ring D, or isomorphic to the ring of a Morita context with zero pairings where both of the underlying rings are division rings. We also classify the rings that consist entirely of units, idempotents, and nilpotent elements. In our study of clean group rings, we show exactly when the group ring Z(p)Cn is clean, where Z(p) is the localization of the integers at p, and Cn is the cyclic group of order n. It is well known that the group ring Z(7)C3 is not clean even though the group ring Z(p)C3 is quasiclean, semiclean, and Σ-clean for any prime p, and 2-clean for any prime p = 2. We prove that Z(p)C3 is clean if and only if p ≢ 1 modulo 3. More generally, we prove that the group ring Z(p)Cn is clean if and only if p is a primitive root of m, where n = pkm and p does not divide m. We also consider the problems of classifying the groups G whose group rings RG are clean for any clean ring R, and of classifying the rings R such that the group ring RG of any locally finite group G o (open full item for complete abstract)

Committee: Warren McGovern Ph.D. (Advisor); Rieuwert Blok Ph.D. (Advisor); Sheila Roberts Ph.D. (Committee Member); Mihai Staic Ph.D. (Committee Member) Subjects: Mathematics