Dissertations / Theses: 'Commutative Rings and Algebras'

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Malec, Sara. "Intersection Algebras and Pointed Rational Cones." Digital Archive @ GSU, 2013. http://digitalarchive.gsu.edu/math_diss/14.

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In this dissertation we study the algebraic properties of the intersection algebra of two ideals I and J in a Noetherian ring R. A major part of the dissertation is devoted to the finite generation of these algebras and developing methods of obtaining their generators when the algebra is finitely generated. We prove that the intersection algebra is a finitely generated R-algebra when R is a Unique Factorization Domain and the two ideals are principal, and use fans of cones to find the algebra generators. This is done in Chapter 2, which concludes with introducing a new class of algebras called fan algebras. Chapter 3 deals with the intersection algebra of principal monomial ideals in a polynomial ring, where the theory of semigroup rings and toric ideals can be used. A detailed investigation of the intersection algebra of the polynomial ring in one variable is obtained. The intersection algebra in this case is connected to semigroup rings associated to systems of linear diophantine equations with integer coefficients, introduced by Stanley. In Chapter 4, we present a method for obtaining the generators of the intersection algebra for arbitrary monomial ideals in the polynomial ring.

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Sekaran, Rajakrishnar. "Fuzzy ideals in commutative rings." Thesis, Rhodes University, 1995. http://hdl.handle.net/10962/d1005221.

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In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.

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Hasse, Erik Gregory. "Lowest terms in commutative rings." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6433.

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Putting fractions in lowest terms is a common problem for basic algebra courses, but it is rarely discussed in abstract algebra. In a 1990 paper, D.D. Anderson, D.F. Anderson, and M. Zafrullah published a paper called Factorization in Integral Domains, which summarized the results concerning different factorization properties in domains. In it, they defined an LT domain as one where every fraction is equal to a fraction in lowest terms. That is, for any x/y in the field of fractions of D, there is some a/b with x/y=a/b and the greatest common divisor of a and b is 1. In addition, R. Gilmer included a brief exercise concerning lowest terms over a domain in his book Multiplicative Ideal Theory. In this thesis, we expand upon those definitions. First, in Chapter 2 we make a distinction between putting a fraction in lowest terms and reducing it to lowest terms. In the first case, we simply require the existence of an equal fraction which is in lowest terms, while the second requires an element which divides both the numerator and the denominator to reach lowest terms. We also define essentially unique lowest terms, which requires a fraction to have only one lowest terms representation up to unit multiples. We prove that a reduced lowest terms domain is equivalent to a weak GCD domain, and that a domain which is both a reduced lowest terms domain and a unique lowest terms domain is equivalent to a GCD domain. We also provide an example showing that not every domain is a lowest terms domain as well as an example showing that putting a fraction in lowest terms is a strictly weaker condition than reducing it to lowest terms. Next, in Chapter 3 we discuss how lowest terms in a domain interacts with the polynomial ring. We prove that if D[T] is a unique lowest terms domain, then D must be a GCD domain. We also provide an alternative approach to some of the earlier results using the group of divisibility. So far, all fractions have been representatives of the field of fractions of a domain. However, in Chapter 4 we examine fractions in other localizations of a domain. We define a necessary and sufficient condition on the multiplicatively closed set, and then examine how this relates to existing properties of multiplicatively closed sets. Finally, in Chapter 5 we briefly examine lowest terms in rings with zero divisors. Because many properties of GCDs do not hold in such rings, this proved difficult. However, we were able to prove some results from Chapter 2 in this more general case.

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Bell, Kathleen. "Cayley Graphs of PSL(2) over Finite Commutative Rings." TopSCHOLAR®, 2018. https://digitalcommons.wku.edu/theses/2102.

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Hadwiger's conjecture is one of the deepest open questions in graph theory, and Cayley graphs are an applicable and useful subtopic of algebra. Chapter 1 will introduce Hadwiger's conjecture and Cayley graphs, providing a summary of background information on those topics, and continuing by introducing our problem. Chapter 2 will provide necessary definitions. Chapter 3 will give a brief survey of background information and of the existing literature on Hadwiger's conjecture, Hamiltonicity, and the isoperimetric number; in this chapter we will explore what cases are already shown and what the most recent results are. Chapter 4 will give our decomposition theorem about PSL2 (R). Chapter 5 will continue with corollaries of the decomposition theorem, including showing that Hadwiger's conjecture holds for our Cayley graphs. Chapter 6 will finish with some interesting examples.

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Granger, Ginger Thibodeaux. "Properties of R-Modules." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc500710/.

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This thesis investigates some of the properties of R-modules. The material is presented in three chapters. Definitions and theorems which are assumed are stated in Chapter I. Proofs of these theorems may be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1958. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Properties of R-modules are developed in Chapter II. The most important results presented in this chapter include existence theorems for R-modules and properties of submodules in R-modules. The third and final chapter presents an example which illustrates how a ring R, may be regarded as an R-module and speaks of the direct sum of ideals of a ring as a direct sum of submodules.

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Oman, Gregory Grant. "A generalization of Jónsson modules over commutative rings with identity." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1164331653.

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Coughlin, Heather. "Classes of normal monomial ideals /." view abstract or download file of text, 2004. http://wwwlib.umi.com/cr/uoregon/fullcit?p3147816//.

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Thesis (Ph. D.)--University of Oregon, 2004.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 85-86). Also available for download via the World Wide Web; free to University of Oregon users.

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Green, Ellen Yvonne. "Characterizing the strong two-generators of certain Noetherian domains." CSUSB ScholarWorks, 1997. https://scholarworks.lib.csusb.edu/etd-project/1539.

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Salt, Brittney M. "MONOID RINGS AND STRONGLY TWO-GENERATED IDEALS." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/31.

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This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.

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Race, Denise T. (Denise Tatsch). "Containment Relations Between Classes of Regular Ideals in a Ring with Few Zero Divisors." Thesis, North Texas State University, 1987. https://digital.library.unt.edu/ark:/67531/metadc331394/.

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This dissertation focuses on the significance of containment relations between the above mentioned classes of ideals. The main problem considered in Chapter II is determining conditions which lead a ring to be a P-ring, D-ring, or AM-ring when every regular ideal is a P-ideal, D-ideal, or AM-ideal, respectively. We also consider containment relations between classes of regular ideals which guarantee that the ring is a quasi-valuation ring. We continue this study into the third chapter; in particular, we look at the conditions in a quasi-valuation ring which lead to a = Jr, sr - f, and a = v. Furthermore we give necessary and sufficient conditions that a ring be a discrete rank one quasi-valuation ring. For example, if R is Noetherian, then ft = J if and only if R is a discrete rank one quasi-valuation ring.

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Oyinsan, Sola. "Primary decomposition of ideals in a ring." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3289.

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The concept of unique factorization was first recognized in the 1840s, but even then, it was still fairly believed to be automatic. The error of this assumption was exposed largely through attempts to prove Pierre de Fermat's, 1601-1665, last theorem. Once mathematicians discovered that this property did not always hold, it was only natural for them to try to search for the strongest available alternative. Thus began the attempt to generalize unique factorization. Using the ascending chain condition on principle ideals, we will show the conditions under which a ring is a unique factorization domain.

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Zagrodny, Christopher Michael. "Algebraic Concepts in the Study of Graphs and Simplicial Complexes." Digital Archive @ GSU, 2006. http://digitalarchive.gsu.edu/math_theses/7.

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This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs.

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Le, Gros Giovanna. "Minimal approximations for cotorsion pairs generated by modules of projective dimension at most one over commutative rings." Doctoral thesis, Università degli studi di Padova, 2019. http://hdl.handle.net/11577/3423180.

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In this thesis we study cotorsion pairs (A, B) generated by classes of R-modules of projective dimension at most one. We are interested in when these cotorsion pairs provide covers or envelopes over commutative rings. More precisely, we investigate Enochs' Conjecture in this setting. That is, for a class A contained in the class of modules of projective dimension at most one, denoted P_1, we investigate the question of whether A is covering necessarily implies that A is closed under direct limits. Additionally, under certain restrictions we characterise the rings which satisfy this property. To this end, there were two cases to consider: when the cotorsion pair is of finite type and when it is not of finite type. For the case that the cotorsion pair (P_1, B) is not (necessarily) of finite type, we show that over a semihereditary ring R, if P_1 is covering it must be closed under direct limits. This gives an example of a cotorsion pair not of finite type which satisfies Enochs' Conjecture. The next part of the thesis is dedicated toward cotorsion pairs of finite type, specifically the 1-tilting cotorsion pairs over commutative rings. We rely heavily on work of Hrbek who characterises these cotorsion pairs over commutative rings, as well as work of Positselski and Bazzoni-Positselski in their work on contramodules. We consider the case of a 1-tilting cotorsion pair (A, T) over a commutative ring with an associated Gabriel topology G, and begin by investigating when T is an enveloping class. We find that if T is enveloping, then the associated Gabriel topology must arise from a perfect localisation. That is, G must arise from a flat ring epimorphism from R to R_G, where R_G is the ring of quotients of R with respect to G. Furthermore, if G arises from a perfect localisation, T is enveloping in Mod-R if and only if the projective dimension of R_G is less than or equal to one and R/J is a perfect ring for every ideal J in G if and only if the projective dimension of R_G is less than or equal to one and the topological ring End(R_G/R) is pro-perfect. Next, we consider the case that A is a covering class, and we prove that A is covering in Mod-R if and only if the projective dimension of R_G is less than or equal to one and both the localisation R_G is a perfect ring and R/J is a perfect ring for every J in G. Additionally, we study general cotorsion pairs, as well as conditions for an approximation to be a minimal approximation. Moreover, we consider a hereditary cotorsion pair and show that if it provides covers it must provide envelopes.
In questa tesi studiamo le coppie di cotorsione (A, B) generate da classi di R-moduli di dimensione proiettiva al più uno. Siamo interessati nel caso in cui queste coppie di cotorsione ammettano ricoprimenti o inviluppi su anelli commutativi. Più precisamente, indaghiamo la congettura di Enochs per A. Cioè, per A contenuta nella classe P_1, che denota la classe di R-moduli di dimensione proiettiva al più uno, cerchiamo di capire se per A una classe ricoprente allora necessariamente implica che A è chiusa per limiti diretti. In più, con certe restrizioni, descriviamo gli anelli che soddisfano questa proprietà. Ci sono due casi da considerare: il caso di coppia di cotorsione di tipo finito e il caso non di tipo finito. Quando la coppia di cotorsione non è (necessariamente) di tipo finito, dimostriamo che per un anello commutativo semiereditario R, se P_1 è una classe ricoprente, deve essere chiusa per limiti diretti. Questo ci da un esempio di una coppia di cotorsione che non è di tipo finito che soddisfa la congettura di Enochs. Successivamente, analizziamo le coppie di cotorsione di tipo finito. Specificamente, le coppie di cotorsione 1-tilting su anelli commutativi. A questo scopo sono indispensabili il lavoro di Hrbek, che caratterizza tali coppie di cotorsione su anelli commutativi, e il lavoro di Positselski e Bazzoni-Positselski nel loro lavoro sui contramoduli. Consideriamo il caso di una coppia di cotorsione 1-tilting (A, T) su un anello commutativo con una topologia di Gabriel associata G, e studiamo quando (A, T) ammette inviluppi. Troviamo che se T ammette inviluppi, G è una topologia di Gabriel perfetta. Cioè, G viene da un epimorfismo piatto di anelli da R a R_G dove R_G è la localizzazione di R rispetto a G. Inoltre, se G è una topologia di Gabriel perfetta, T ammette inviluppi se e solo se R_G ha dimensione proiettiva al più uno e R/J è un anello perfetto per tutti gli ideali J in G se e solo se R_G ha dimensione proiettiva al più uno e l'anello topologico End(R_G/R) è pro-perfetto. Poi consideriamo il caso in cui A è ricoprente. Dimostriamo che A è ricoprente in Mod-R se e solo se R_G ha dimensione proiettiva al più uno e R_G è un anello perfetto e R/J è perfetto per ogni J in G. In aggiunta, studiamo coppie di cotorsione in generale e studiamo condizioni sufficienti affinchè una approssimazione sia minimale. Inoltre, consideriamo una coppia di cotorsione ereditaria e dimostriamo che se ammette ricoprimenti deve ammettere inviluppi.

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Byun, Eui Won James. "Affine varieties, Groebner basis, and applications." CSUSB ScholarWorks, 2000. https://scholarworks.lib.csusb.edu/etd-project/1611.

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Diaz, Noguera Maribel del Carmen. "Sobre derivações localmente nilpotentes dos aneis K[x,y,z] e K[x,y]." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306307.

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Orientador: Paulo Roberto Brumatti
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Ciencia da Computação
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Resumo: O principal objetivo desta dissertação é apresentar resultados centrais sobre derivações localmente nilpotentes no anel de polinômios B = k[x1, ..., xn], para n = 3 que foram apresentados por Daniel Daigle em [2 ], [3] e [4] .Para este propósito, introduziremos os conceitos básicos e fundamentais da teoria das derivações num anel e apresentaremos resultados em relação a derivações localmente nilpotentes num domínio de característica zero e de fatorização única. Entre tais resultados está a fórmula Jacobiana que usaremos para descrever o conjunto das derivações equivalentes e localmente nilpotentes de B = k[x, y, z] e o conjunto LND(B), com B = k[x,y]. Também, explicítam-se condições equivalentes para a existência de uma derivação ?-homogênea e localmente nilpotente de B = k[x, y, z] com núcleo k[¿, g], onde {¿}, {g} e B, mdc(?) = mdc(?(¿), ? (g)) = 1
Abstract: In this dissertation we present centraIs results on locally nilpotents derivations in a ring of polynomials B = k[x1, ..., xn], for n = 3, which were presented by Daniel Daigle in [2], [3] and [4]. For this, we introduce basic fundamenta1 results of the theory of derivations in a ring and we present results on locally nilpotents derivations in a domain with characteristic zero and unique factorization. One of these results is the Jacobian forrnula that we use to describe the set of the equivalent loca11y nilpotents derivations of B = k[x, y, z] and the set LND(B) where B = k[x, y]. Moreover, we give equivalent conditions to the existence of a ?-homogeneous locally nilpotent derivation in the ring B = k[x, y, z] with kernel k[¿, g], {¿} and {g} e B, and mdc(?) = mdc(?(¿), ? (g)) = 1
Mestrado
Algebra
Mestre em Matemática

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Berni, Jean Cerqueira. "Some algebraic and logical aspects of C&#8734-Rings." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-14022019-203839/.

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As pointed out by I. Moerdijk and G. Reyes in [63], C∞-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C∞-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C∞-rings. Next we develop some topics of what we call a ∞Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C∞-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C∞-rings, such as ∞(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of ∞ rings, the (coherent) theory of local C∞-rings and the (algebraic) theory of von Neumann regular C∞-rings.
Conforme observado por I. Moerdijk e G. Reyes em [63], os anéis C∞ têm sido estudados especialmente tendo em vista suas aplicações em Teoria de Singularidades e para construir toposes que sirvam de modelos para a Geometria Diferencial Sintética. Neste trabalho, seguimos um caminho complementar, aprofundando nosso conhecimento sobre eles por um viés mais puro, fazendo uso da Teoria das Categorias e os analisando a partir de pontos de vista algébrico e lógico-categorial. Iniciamos o trabalho apresentando uma sistematização abrangente dos fatos fundamentais da teoria (equacional) dos anéis C∞, distribuídos aqui e ali na literatura atual - a maioria sem demonstrações - mas que servem de base para a teoria. Na sequência, desenvolvemos alguns tópicos do que denominamos Álgebra Comutativa C∞, expandindo resultados parciais de [66] e [67]. Realizamos um estudo sistemático dos anéis C∞ von Neumann-regulares - na linha do estudo algébrico realizado em [2]- e apresentamos alguns resultados interessantes a seu respeito, juntamente com sua relação (funtorial) com os espaços booleanos. Estudamos algumas noções pertinentes à Teoria de Feixes para anéis ∞, tais como espaços (localmente) ∞anelados e o sítio de Zariski liso. Finalmente, descrevemos toposes classicantes para a teoria (algébrica) dos anéis C∞, a teoria (coerente) dos anéis locais C∞ e a teoria (algébrica) dos anéis C∞ von Neumann regulares.

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Ferreira, Mauricio de Araujo 1982. "Algebras biquaternionicas : construção, classificação e condições de existencia via formas quadraticas e involuções." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306541.

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Orientador: Antonio Jose Engler
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho, estudamos as álgebras biquaterniônicas, que são um tipo especial de álgebra central simples de dimensão 16, obtida como produto tensorial de duas álgebras de quatérnios. A teoria de formas quadráticas é aplicada para estudarmos critérios de decisão sobre quando uma álgebra biquaterniônica é de divisão e quando duas destas álgebras são isomorfas. Além disso, utilizamos o u-invariante do corpo para discutirmos a existência de álgebras biquaterniônicas de divisão sobre o corpo. Provamos também um resultado atribuído a A. A. Albert, que estabelece critérios para decidir quando uma álgebra central simples de dimensão 16 é de fato uma álgebra biquaterniônica, através do estudo de involuções. Ao longo do trabalho, construímos vários exemplos concretos de álgebras biquaterniônicas satisfazendo propriedades importantes
Mestrado
Algebra
Mestre em Matemática

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Gokhale, Dhananjay R. "Resolutions mod I, Golod pairs." Diss., Virginia Tech, 1992. http://hdl.handle.net/10919/39431.

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Let R be a commutative ring, I be an ideal in R and let M be a R/ I -module. In this thesis we construct a R/ I -projective resolution of M using given R-projective resolutions of M and I. As immediate consequences of our construction we give descriptions of the canonical maps Ext_R/I(M,N) -> Ext_R(M,N) and Tor^R_N(M, N) -> Tor^R/I_n(M, N) for a R/I module N and we give a new proof of a theorem of Gulliksen [6] which states that if I is generated by a regular sequence of length r then ∐∞_n=o Tor^R/I_n (M, N) is a graded module over the polynomial ring R/ I [X₁. .. X_r] with deg X_i = -2, 1 ≤ i ≤ r. If I is generated by a regular element and if the R-projective dimension of M is finite, we show that M has a R/ I-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when R = (R, m) is a Noetherian local ring and M is a finitely generated R/ I -module, we discuss the minimality of the constructed resolution. If it is minimal we call (M, I) a Golod pair over R. We give a direct proof of a theorem of Levin [10] which states thdt if (M,I) is a Golod pair over R then (Ωⁿ_R/IR/I(M),I) is a Golod pair over R where Ωⁿ_R/IR/I(M) is the nth syzygy of the constructed R/ I -projective resolution of M. We show that the converse of the last theorem is not true and if (Ω¹_R/IR/I(M),I) is a Golod pair over R then we give a necessary and sufficient condition for (M, I) to be a Golod pair over R. Finally we prove that if (M, I) is a Golod pair over R and if a ∈ I - mI is a regular element in R then (M, (a)) and (1/(a), (a)) are Golod pairs over R and (M,I/(a)) is a Golod pair over R/(a). As a corrolary of this result we show that if the natural map π : R → R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin [8]), then the natural maps π₁ : R → R/(a) and π₂ : R/(a) → R/1 are Golod homomorphisms.
Ph. D.

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Lavila, Vidal Olga. "On the diagonals of a Rees algebra." Doctoral thesis, Universitat de Barcelona, 1999. http://hdl.handle.net/10803/53578.

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The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogenous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. First we are going to introduce the subject and the main problems. After that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work.
L’objectiu d’aquesta memòria és l’estudi de les propietats aritmètiques de les diagonals d’una àlgebra de Rees o, des d’un punt de vista geomètric, dels anells de coordenades homogenis d’immersions d’explosions de varietats projectives al llarg d’una subvarietat. En primer lloc, anem a introduir el tema i els principals problemes que tractarem. A continuació, exposarem els resultats coneguts sobre aquests problemes i finalment farem un resum dels resultats obtinguts en aquesta memòria.

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Johnston, Ann. "Markov Bases for Noncommutative Harmonic Analysis of Partially Ranked Data." Scholarship @ Claremont, 2011. http://scholarship.claremont.edu/hmc_theses/4.

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Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool of algebraic statistics, particularly for the study of fully ranked data. In this thesis, we explore the extension of this technique for data analysis to the study of partially ranked data, focusing on data from surveys in which participants are asked to identify their top $k$ choices of $n$ items. Before we move on to our own data analysis, though, we present a thorough discussion of the Diaconis–Sturmfels algorithm and its use in data analysis. In this discussion, we attempt to collect together all of the background on Markov bases, Markov proceses, Gröbner bases, implicitization theory, and elimination theory, that is necessary for a full understanding of this approach to data analysis.

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Margolin, Benjamin Paul. "Non-commutative deformation rings." Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/3135.

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The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem. The thesis is divided into two parts. In the first part, A is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field k. A category Ĉ is defined whose objects are complete local λ-algebras R with residue field k such that R is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If V is a finite dimensional k-vector space that is also a left A-module and that satisfies a natural finiteness condition, it is proved that V has a so-called versal deformation ring R(A,V). More precisely, R(A,V) is an object in Ĉ such that the isomorphism class of every lift of V over an object R in Ĉ arises from a morphism α : R(A,V)→ R in Ĉ and α is unique if R is the ring of dual numbers k[ϵ]. In the second part, two particular examples of λ, A and V are studied and the versal deformation ring R(A,V) is determined in each of these cases. In the first example, λ=k, A is a series of non-commutative k-algebras depending on a parameter r≥2, and V is a particular quotient module of A; it is shown that R(A,V) is isomorphic to A. The second example builds on the first example when r=2 and uses that, if additionally the characteristic of k is 2, then A is isomorphic to the group ring k[D8] of a dihedral group D8 of order 8. It is shown that if k is perfect and W is the ring of infinite Witt vectors over k, then R(W[D8],V) is isomorphic to W[D8].

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22

Duncan, A. J. "Two topics in commutative ring theory." Thesis, University of Edinburgh, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234124.

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Choi, Yemon. "Cohomology of commutative Banach algebras and lÂ¹-semigroup algebras." Thesis, University of Newcastle Upon Tyne, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.427291.

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Akalan, Evrim. "Generalizations of non-commutative uniquefactorization rings." Thesis, University of Warwick, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.502119.

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There is a well-developed theory of unique factorization domains in commutative algebra. The generalization of this concept to non-commutative rings has also been extensively studied (e.g. in [19], [16], [1]). This thesis is concerned with classes of non-commutative rings which are generalizations of non-commutative Noetherian unique factorization rings. Noetherian UFR's are maximal orders and every reflexive ideal is invertible in these rings. Clearly maximal orders and reflexive ideals are important concepts and we examine them in this thesis. UFR's can also be considered as rings in which the divisor class group is trivial. This provides the motivation for us to study this group more generally. In Chapter 1, we give the basic material we shall need from the theory of noncommutative rings-and in Chapter 2, we present the known results about certain classes of rings which are crucial for this thesis. Chapters 3, 4 and 5 contain the original work of this thesis. In Chapter 3, we study the prime Noetherian maximal orders with enough invertible ideals. We show that in such rings every height 1 prime ideal is maximal reflexive and we prove results which generalize some of the results of Asano orders. In Chapter 4, we investigate divisor class groups. We also study the relations between these groups and the divisor -class group of the centre of the ring. In Chapter 5, we introduce the Generalized Dedekind prime rings (G-Dedekind prime rings), which are also a generalization of Noetherian UFR's, and we study this class of rings.

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25

Vo, Monika. "New classes of finite commutative rings." Thesis, University of Hawaii at Manoa, 2003. http://proquest.umi.com/pqdweb?index=2&did=765961151&SrchMode=1&sid=1&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1208558919&clientId=23440.

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Moore, David. "Endomorphisms of commutative unital Banach algebras." Thesis, University of Nottingham, 2017. http://eprints.nottingham.ac.uk/39674/.

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This thesis is a collection of theorems which say something about the following question: if we know that a bounded operator on a commutative unital Banach algebra is a unital endomorphism, what can we say about its other properties? More specifically, the majority of results say something about how the spectrum of a commutative unital Banach algebra endomorphism is dependent upon the properties of the algebra on which it acts. The main result of the thesis (the subject of Chapter 3) reveals that primary ideals (that is, ideals with single point hulls) can sometimes be particularly important in questions of this type. The thesis also contains some contributions to the Fredholm theory for bounded operators on an arbitrary complex Banach space. The second major result of the thesis is in this direction, and concerns the relationship between the essential spectrum of a bounded operator on a Banach space and those of its restrictions and quotients - `to' and `by' - closed invariant subspaces.

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Dawson, Thomas. "Extensions of normed algebras." Thesis, University of Nottingham, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.288997.

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Rumbos, Irma Beatriz. "A sheaf representation for non-commutative rings /." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=70356.

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For any ring R (associative with 1) we associate a space X of prime torsion theories endowed with Golan's SBO-topology. A separated presheaf L(-,M) on X is then constructed for any right R-module M$ sb{ rm R}$, and a sufficient condition on M is given such that L(-,M) is actually a sheaf. The sheaf space rm E { buildrel{ rm p} over longrightarrow} X) etermined by L(-,M) represents M in the following sense: M is isomorphic to the module of continuous global sections of p. These results are applied to the right R-module R$ sb{ rm R}$ and it is seen that semiprime rings satisfy the required condition for L(-,R) to be a sheaf. Among semiprime rings two classes are singled out, fully symmetric semiprime and right noetherian semiprime rings; these two kinds of rings have the desirable property of yielding "nice" stalks for the above sheaf.

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29

Denizler, Ismail Hakki. "Artinian modules over commutative rings, and duality." Thesis, University of Sheffield, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.364185.

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30

Zuick, Nhan H. "The Gelfand Theorem for Commutative Banach Algebras." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/243.

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We give an overview of the basic properties of Banach Algebras. After that we specialize to the case of commutative Banach Algebras and study the Gelfand Map. We study the main characteristic of that map, and work on some applications.

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Archer, Louise. "Hall algebras and Green rings." Thesis, University of Oxford, 2005. http://ora.ox.ac.uk/objects/uuid:960af4b3-8f32-4263-9142-261f49d52405.

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This thesis consists of two parts, both of which involve the study of algebraic structures constructed via the multiplication of modules. In the first part we look at Hall algebras. We consider the Hall algebra of a cyclic quiver algebra with relations of length two and present a multiplication formula for the explicit calculation of products in this algebra. We then look at the case of a cyclic quiver with two vertices and describe the corresponding composition algebra as a quotient of the positive part of a quantised enveloping algebra of type Ã₁ We then look at quotients of Hall algebras of self-injective algebras. We give an abstract result describing the quotient of such a Hall algebra by the ideal generated by isomorphism classes of projective modules, and also a more explicit result describing quotients of Hall algebras of group algebras for cyclic 2-groups and some related polynomial algebras. The second part of the thesis deals with Green rings. We compare the Green rings of a group algebra and the corresponding Jennings algebra for certain p-groups. It is shown that these two Green rings are isomorphic in the case of a cyclic p-group. In the case of the Klein four group it is shown that the two Green rings are not isomorphic, but that there exist quotients of these rings which are isomorphic. It is conjectured that the corresponding quotients will also be isomorphic in the case of a dihedral 2-group. The properties of these quotients are studied, with the aim of producing evidence to support this conjecture. The work on Green rings also includes some results on the realisation of quotients of Green rings as group rings over ℤ.

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at, Peter Michor@esi ac. "Smooth $*$--Algebras." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1046.ps.

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33

Mooney, Christopher Park. "Generalized factorization in commutative rings with zero-divisors." Thesis, The University of Iowa, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3595128.

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The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of τ-factorization, studied extensively by A. Frazier and D.D. Anderson.

Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements.

In this thesis, we investigate several methods for extending the theory of τ-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. Agˇargün and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations.

This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using τ_z-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using τ-U-factorization, we are able to answer many questions that arise when discussing direct products of rings.

There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending τ-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.

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Coutinho, S. C. "Generating modules efficiently over non-commutative noetherian rings." Thesis, University of Leeds, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372589.

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35

Hayes, Leslie Danielle. "The plus closure of an ideal." Access restricted to users with UT Austin EID Full text (PDF) from UMI/Dissertation Abstracts International, 2001. http://wwwlib.umi.com/cr/utexas/fullcit?p3035948.

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36

Gawell, Elin. "Centra of Quiver Algebras." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-106734.

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A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiveralgebra and state necessary and sufficient conditions for the center to be finitely genteratedas a K-algebra.Examples are provided of partly (anti-)commutative quiver algebras that are Koszul algebras. Necessary and sufficient conditions for finite generation of the Hochschild cohomology ring modulo nilpotent elements for a partly (anti-)commutative Koszul quiver algebra are given.

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37

Hart, Robert. "A Non-commutative *-algebra of Borel Functions." Thèse, Université d'Ottawa / University of Ottawa, 2012. http://hdl.handle.net/10393/23235.

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To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

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Volkweis, Leite Samuel [Verfasser]. "p-adic Representations of Commutative Rings / Samuel Volkweis Leite." Konstanz : Bibliothek der Universität Konstanz, 2013. http://d-nb.info/1031879943/34.

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39

Emerson, Sharon Sue. "Overrings of an Integral Domain." Thesis, University of North Texas, 1992. https://digital.library.unt.edu/ark:/67531/metadc332679/.

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This dissertation focuses on the properties of a domain which has the property that each ideal is a finite intersection of a π-ideal, the properties of a domain which have the property that each ideal is a finite product of π-ideal, and the containment relations of the resulting classes of ideals. Chapter 1 states definitions which are needed in later chapters. Chapters 2 and 3 focuses on domains which have the property that each ideal in D is a finite intersection of π-ideals while Chapter 4 focuses on domains with the property that each ideal is a finite product of π-ideals. Chapter 5 discusses the containment relations which occur as a result of Chapters 2 and 3.

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40

Nenashev, Gleb. "On a class of commutative algebras associated to graphs." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-132987.

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In 2004 Alexander Postnikov and Boris Shapiro introduced a class of commutative algebras for non-directed graphs. There are two main types of such algebras, algebras of the first type count spanning trees and algebras of the second type count spanning forests. These algebras have a number of interesting properties including an explicit formula for their Hilbert series. In this thesis we mainly work with the second type of algebras, we discover more properties of the original algebra and construct a few generalizations. In particular we prove that the algebra counting forests depends only on graphical matroid of the graph and converse. Furthermore, its "K-theoretic" filtration reconstructs the whole graph. We introduse $t$ labelled algebras of a graph, their Hilbert series contains complete information about the Tutte polynomial of the initial graph. Finally we introduce similar algebras for hypergraphs. To do this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid.

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41

Chattopadhyay, Arkadev. "Circuits, communication and polynomials." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=115660.

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In this thesis, we prove unconditional lower bounds on resources needed to compute explicit functions in the following three models of computation: constant-depth boolean circuits, multivariate polynomials over commutative rings and the 'Number on the Forehead' model of multiparty communication. Apart from using tools from diverse areas, we exploit the rich interplay between these models to make progress on questions arising in the study of each of them.
Boolean circuits are natural computing devices and are ubiquitous in the modern electronic age. We study the limitation of this model when the depth of circuits is fixed, independent of the length of the input. The power of such constant-depth circuits using gates computing modular counting functions remains undetermined, despite intensive efforts for nearly twenty years. We make progress on two fronts: let m be a number having r distinct prime factors none of which divides ℓ. We first show that constant depth circuits employing AND/OR/MODm gates cannot compute efficiently the MAJORITY and MODℓ function on n bits if 'few' MODm gates are allowed, i.e. they need size nW&parl0;1s&parl0;log n&parr0;1/&parl0;r-1&parr0;&parr0; if s MODm gates are allowed in the circuit. Second, we analyze circuits that comprise only MOD m gates, We show that in sub-linear size (and arbitrary depth), they cannot compute AND of n bits. Further, we establish that in that size they can only very poorly approximate MODℓ.
Our first result on circuits is derived by introducing a novel notion of computation of boolean functions by polynomials. The study of degree as a resource in polynomial representation of boolean functions is of much independent interest. Our notion, called the weak generalized representation, generalizes all previously studied notions of computation by polynomials over finite commutative rings. We prove that over the ring Zm , polynomials need Wlogn 1/r-1 degree to represent, in our sense, simple functions like MAJORITY and MODℓ. Using ideas from arguments in communication complexity, we simplify and strengthen the breakthrough work of Bourgain showing that functions computed by o(log n)-degree polynomials over Zm do not even correlate well with MODℓ.
Finally, we study the 'Number on the Forehead' model of multiparty communication that was introduced by Chandra, Furst and Lipton [CFL83]. We obtain fresh insight into this model by studying the class CCk of languages that have constant k-party deterministic communication complexity under every possible partition of input bits among parties. This study is motivated by Szegedy's [Sze93] surprising result that languages in CC2 can all be extremely efficiently recognized by very shallow boolean circuits. In contrast, we show that even CC 3 contains languages of arbitrarily large circuit complexity. On the other hand, we show that the advantage of multiple players over two players is significantly curtailed for computing two simple classes of languages: languages that have a neutral letter and those that are symmetric.
Extending the recent breakthrough works of Sherstov [She07, She08b] for two-party communication, we prove strong lower bounds on multiparty communication complexity of functions. First, we obtain a bound of n O(1) on the k-party randomized communication complexity of a function that is computable by constant-depth circuits using AND/OR gates, when k is a constant. The bound holds as long as protocols are required to have better than inverse exponential (i.e. 2-no1 ) advantage over random guessing. This is strong enough to yield lower bounds on the size of an important class of depth-three circuits: circuits having a MAJORITY gate at its output, a middle layer of gates computing arbitrary symmetric functions and a base layer of arbitrary gates of restricted fan-in.
Second, we obtain nO(1) lower bounds on the k-party randomized (bounded error) communication complexity of the Disjointness function. This resolves a major open question in multiparty communication complexity with applications to proof complexity. Our techniques in obtaining the last two bounds, exploit connections between representation by polynomials over teals of a boolean function and communication complexity of a closely related function.

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42

Weaver, Martha Ellen. "Graded artin algebras, coverings and factor rings." Diss., Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/82612.

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Let (Γ,ρ) be a directed graph with relations. Let F: Γ’ → Γ be a topological covering. It is proved in this thesis that there is a set of relations ρ̅ on Γ such that the category of K-respresentations of Γ’ whose images under the covering functor satisfy ρ is equivalent to the category of finite-dimensional, grades KΓ/<ρ̅>-modules. If Γ’ is the universal cover of Γ, then this category is called the category of unwindable KΓ/<ρ>-modules. For arrow unique graphs it is shown that the category of unwindable KΓ/<ρ>-modules does not depend on <ρ>. Also, it is shown that for arrow unique graphs the finite dimensional uniserial KΓ/<ρ>-modules are unwindable. Let Γ be an arrow unique graph with commutativity relations, ρ. In Section 2, the concept of unwindable modules is used to determine whether a certain factor ring of KΓ/<ρ> is of finite representation type. In a different vein, the relationship between almost split sequences over Artin algebras and the almost split sequences over factor rings of such algebras is studied. Let Λ be an Artin algebra and let Λ̅ be a factor ring of Λ. Two sets of necessary and sufficient conditions are obtained for determining when an almost split sequence of Λ̅-modules remains an almost split sequence when viewed as a sequence of Λ-modules.
Ph. D.

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43

Trentham, Stacy Michelle. "Atomicity in Rings with Zero Divisors." Diss., North Dakota State University, 2011. https://hdl.handle.net/10365/28905.

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In this dissertation, we examine atomicity in rings with zero divisions. We begin by examining the relationship between a ring’s level of atomicity and the highest level of irreducibility shared by the ring’s irreducible elements. Later, we chose one of the higher forms of atomicity and identify ways of building large classes of examples of rings that rise to this level of atomicity but no higher. Characteristics of the various types of irreducible elements will also be examined. Next, we extend our view to include polynomial extensions of rings with zero divisors. In particular, we focus on properties of the three forms of maximal common divisors and how a ring’s classification as an MCD, SMCD, or VSMCD ring affects its atomicity. To conclude, we identify some unsolved problems relating to the topics discussed in this dissertation.

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44

Bradford, Jeremy. "Commutative endomorphism rings of simple abelian varieties over finite fields." Thesis, University of Maryland, College Park, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3557641.

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In this thesis we look at simple abelian varieties defined over a finite field k = [special characters omitted]_pn with End_k( A) commutative. We derive a formula that connects the p -rank r(A) with the splitting behavior of p in E = [special characters omitted](π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of p in [special characters omitted]_E, and we do so for abelian varieties of dimension less than or equal to four defined over [special characters omitted]_p. We then look for when p divides [[special characters omitted]_E : [special characters omitted][π, π]]. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p ≥ 3. We also derive a condition that guarantees that p divides [[special characters omitted]_E: [special characters omitted][π, π]]. Last, we explicitly describe the structure of some intermediate subrings of p-power index between [special characters omitted][π, π] and [special characters omitted]_E when A is an abelian 3-fold with r(A) = 1.

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45

Oman, Gregory Grant. "A generalization of Jónsson modules over commutative rings with identity." The Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=osu1164331653.

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46

Gjerling, Andreas. "On rings of quotients of soluble group algebras." Thesis, Queen Mary, University of London, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.286813.

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Quah, James. "Nilpotent elements in green rings of Hopf algebras." Thesis, University of Exeter, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.337805.

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Chasen, Lee A. "The cohomology rings of classical Brauer tree algebras." Diss., Virginia Tech, 1995. http://hdl.handle.net/10919/38572.

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49

Hieta-aho, Erik. "On Finite Rings, Algebras, and Error-Correcting Codes." Ohio University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1525182104493243.

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50

Nordstrom, Hans Erik. "Associated primes over Ore extensions and generalized Weyl algebras /." view abstract or download file of text, 2005. http://wwwlib.umi.com/cr/uoregon/fullcit?p3181118.

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Thesis (Ph. D.)--University of Oregon, 2005.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 48-49). Also available for download via the World Wide Web; free to University of Oregon users.

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Dissertations / Theses on the topic 'Commutative Rings and Algebras'

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