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1
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.
2
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 EnglerDissertação (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
3
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.
4
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.
5
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.
6
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.
7
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.
8
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.
9
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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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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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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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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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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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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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.
16
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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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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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 BrumattiDissertação (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
19
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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Berni, Jean Cerqueira. "Some algebraic and logical aspects of C∞-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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Crawford, Simon Philip. "Singularities of noncommutative surfaces". Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31543.
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The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.
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Martini, Lorenzo. "Local coherence of hearts in the derived category of a commutative ring". Doctoral thesis, Università degli studi di Trento, 2022. http://hdl.handle.net/11572/354322.
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Approximation theory is a fundamental tool in order to study the representation theory of a ring R. Roughly speaking, it consists in determining suitable additive or abelian subcategories of the whole module category Mod-R with nice enough functorial properties. For example, torsion theory is a well suited incarnation of approximation theory. Of course, such an idea has been generalised to the additive setting itself, so that both Mod-R and other interesting categories related with R may be linked functorially. By the seminal work of Beilinson, Bernstein and Deligne (1982), the derived category of the ring turns out to admit useful torsion theories, called t-structures: they are pairs of full subcategories of D(R) whose intersection, called the heart, is always an abelian category. The so-called standard t-structure of D(R) has as its heart the module category Mod-R itself. Since then a lot of results devoted to the module theoretic characterisation of the hearts have been achieved, providing evidence of the usefulness of the t-structures in the representation theory of R. In 2020, following a research line promoted by many other authors, Saorin and Stovicek proved that the heart of any compactly generated t-structure is always a locally finitely presented Grothendieck categories (actually, this is true for any t-structure in a triangulated category with coproducts). Essentially, this means that the hearts of D(R) come equipped with a finiteness condition miming that one valid in Mod-R. In the present thesis we tackle the problem of characterising when the hearts of certain compactly generated t-structures of a commutative ring are even locally coherent. In this commutative context, after the works of Neeman and Alonso, Jeremias and Saorin, compactly generated t-structures turned out to be very interesting over a noetherian ring, for they are in bijection with the Thomason filtrations of the prime spectrum. In other words, they are classified by geometric objects, moreover their constituent subcategories have a precise cohomological description. However, if the ascending chain condition lacks, such classification is somehow partial, though provided by Hrbek. The crucial point is that the constituents of the t-structures have a different description w.r.t. that available in the noetherian setting, yet if one copies the latter for an arbitrary ring still obtains a t-structure, but it is not clear whether it must be compactly generated. Consequently, pursuing the study of the local coherence of the hearts given by a Thomason filtration, we ended by considering two t-structures. Our technique in order to face the lack of the ascending chain condition relies on a further approximation of the hearts by means of suitable torsion theories. The main results of the thesis are the following: we prove that for the so-called weakly bounded below Thomason filtrations the two t-structures have the same heart (therefore it is always locally finitely presented), and we show that they coincide if and only they are both compactly generated. Moreover, we achieve a complete characterisation of the local coherence for the hearts of the Thomason filtrations of finite length.
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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 ExtR/I(M,N) -> ExtR(M,N) and TorRN(M, N) -> TorR/In(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 TorR/In (M, N) is a graded module over the polynomial ring R/ I [X₁. .. Xr] with deg Xi = -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 (ΩnR/IR/I(M),I) is a Golod pair over R where ΩnR/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 (Ω¹R/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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Mbirika, Abukuse III. "Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties". Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/708.
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Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties.
Let R be the polynomial ring Ζ[x1,…,xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal I_h generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for Jh, we identify a basis for the quotient R/Jh.
We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h>h', we have Ih ⊂ Ih' and Jh ⊂ Jh'. We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety.
The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator X and a Hessenberg function h. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as h varies, the quotient R/Jh may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.
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Steward, Michael. "Extending the Skolem Property". The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1492517341492202.
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Di, Lorenzo Andrea. "Integral Chow ring and cohomological invariants of stacks of hyperelliptic curves". Doctoral thesis, Scuola Normale Superiore, 2019. http://hdl.handle.net/11384/85744.
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Ugolini, Matteo. "K3 surfaces". Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18774/.
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Tête, Claire. "Profondeur, dimension et résolutions en algèbre commutative : quelques aspects effectifs". Thesis, Poitiers, 2014. http://www.theses.fr/2014POIT2288/document.
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Cette thèse d'algèbre commutative porte principalement sur la théorie de la profondeur. Nous nous efforçons d'en fournir une approche épurée d'hypothèse noethérienne dans l'espoir d'échapper aux idéaux premiers et ceci afin de manier des objets élémentaires et explicites. Parmi ces objets, figurent les complexes algébriques de Koszul et de Cech dont nous étudions les propriétés cohomologiques grâce à des résultats simples portant sur la cohomologie du totalisé d'un bicomplexe. Dans le cadre de la cohomologie de Cech, nous avons établi la longue suite exacte de Mayer-Vietoris avec un traitement reposant uniquement sur le maniement des éléments. Une autre notion importante est celle de dimension de Krull. Sa caractérisation en termes de monoïdes bords permet de montrer de manière expéditive le théorème d'annulation de Grothendieck en cohomologie de Cech. Nous fournissons également un algorithme permettant de compléter un polynôme homogène en un h.s.o.p.. La profondeur est intimement liée à la théorie des résolutions libres/projectives finies, en témoigne le théorème de Ferrand-Vasconcelos dont nous rapportons une généralisation due à Jouanolou. Par ailleurs, nous revenons sur des résultats faisant intervenir la profondeur des idéaux caractéristiques d'une résolution libre finie. Nous revisitons, dans un cas particulier, une construction due à Tate permettant d'expliciter une résolution projective totalement effective de l'idéal d'un point lisse d'une hypersurface. Enfin, nous abordons la théorie de la régularité en dimension 1 via l'étude des idéaux inversibles et fournissons un algorithme implémenté en Magma calculant l'anneau des entiers d'un corps de nombresThis Commutative Algebra thesis focuses mainly on the depth theory. We try to provide an approach without noetherian hypothesis in order to escape prime ideals and to handle only basic and explicit concepts. We study the algebraic complexes of Koszul and Cech and their cohomological properties by using simple results on the cohomology of the totalization of a bicomplex. In the Cech cohomology context we established the long exact sequence of Mayer-Vietoris only with a treatment based on the elements. Another important concept is that of Krull dimension. Its characterization in terms of monoids allows us to show expeditiously the vanishing Grothendieck theorem in Cech cohomology.We also provide an algorithm to complete a omogeneous polynomial in a h.s.o.p.. The depth is closely related to the theory of finite free/projective resolutions. We report a generalization of the Ferrand-Vasconcelos theorem due to Jouanolou. In addition, we review some results involving the depth of the ideals of expected ranks in a finite free resolution.We revisit, in a particular case, a construction due to Tate. This allows us to give an effective projective resolution of the ideal of a point of a smooth hypersurface. Finally, we discuss the regularity theory in dimension 1 by studying invertible ideals and provide an algorithm implemented in Magma computing the ring of integers of a number field
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"Results on algebraic structures: A-algebras, semigroups and semigroup rings". 1998. http://library.cuhk.edu.hk/record=b6073113.
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by Chen Yuqun.Thesis (Ph.D.)--Chinese University of Hong Kong, 1998.
Includes bibliographical references and index.
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.
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Cecil, Anthony John. "Lie isomorphisms of triangular and block-triangular matrix algebras over commutative rings". Thesis, 2016. http://hdl.handle.net/1828/7471.
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For many matrix algebras, every associative automorphism is inner. We discuss results by Đoković that a non-associative Lie automorphism φ of a triangular matrix algebra Tₙ over a connected unital commutative ring, is of the form φ(A)=SAS⁻¹ + τ(A)I or φ(A)=−SJ Aᵀ JS⁻¹ + τ(A)I, where S ∈ Tₙ is invertible, J is an antidiagonal permutation matrix, and τ is a generalized trace. We incorporate additional arguments by Cao that extended Đoković’s result to unital commutative rings containing nontrivial idempotents.
Following this we develop new results for Lie isomorphisms of block upper-triangular matrix algebras over unique factorization domains. We build on an approach used by Marcoux and Sourour to characterize Lie isomorphisms of nest algebras over separable Hilbert spaces.
We find that these Lie isomorphisms generally follow the form φ = σ + τ where σ is either an associative isomorphism or the negative of an associative anti-isomorphism, and τ is an additive mapping into the center, which maps commutators to zero. This echoes established results by Martindale for simple and prime rings.Graduate
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Khatkar, Meenu. "Units, idempotents, and ideals in certain algebras over commutative rings". Thesis, 2018. http://eprint.iitd.ac.in:80//handle/2074/7991.
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LaGrange, John D. "Zero-Divisor Graphs, Commutative Rings of Quotients, and Boolean Algebras". 2008. http://trace.tennessee.edu/utk_graddiss/393.
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The zero-divisor graph of a commutative ring is the graph whose vertices are the nonzero zero-divisors of the ring such that distinct vertices are adjacent if and only if their product is zero. We use this construction to study the interplay between ring-theoretic and graph-theoretic properties. Of particular interest are Boolean rings and commutative rings of quotients.
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Tlharesakgosi, Batsile. "Topics on z-ideals of commutative rings". Diss., 2017. http://hdl.handle.net/10500/23619.
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The first few chapters of the dissertation will catalogue what is known regarding z-ideals in
commutative rings with identity. Some special attention will be paid to z-ideals in function
rings to show how the presence of the topological description simplifies z-covers of arbitrary
ideals. Conditions in an f-ring that ensure that the sum of z-ideals is a z-ideal will be given.
In the latter part of the dissertation I will generalise a result in higher order z-ideals and
introduce a notion of higher order d-idealsMathematical Sciences
M. Sc. (Mathematics)
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Macoosh, Ruth Rebekka. "Algebraic closures for commutative rings". Thesis, 1987. http://spectrum.library.concordia.ca/4791/1/ML35523.pdf.
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"Theory of distributive modules and related topics". Chinese University of Hong Kong, 1992. http://library.cuhk.edu.hk/record=b5887044.
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by Ng Siu-Hung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1992.
Includes bibliographical references (leaves 80-81).
Introduction --- p.iii
Chapter 1 --- Distributive Modules --- p.1
Chapter 1.1 --- Basic Definitions --- p.1
Chapter 1.2 --- Distributive modules --- p.3
Chapter 1.3 --- Direct sum of distributive modules --- p.9
Chapter 1.4 --- Endomorphisms of a distributive module --- p.13
Chapter 1.5 --- Distributive modules satisfying chain conditions --- p.20
Chapter 2 --- Rings with distributive lattices of right ideals --- p.25
Chapter 2.1 --- Rings of quotients of right D-rings --- p.25
Chapter 2.2 --- Localization of right D-rings --- p.28
Chapter 2.3 --- Reduced primary factorizations in right ND-rings --- p.31
Chapter 2.4 --- ND-rings --- p.38
Chapter 3 --- Distributive modules over commutative rings --- p.43
Chapter 3.1 --- Multiplication modules --- p.43
Chapter 3.2 --- Properties of distributive modules over commutative rings --- p.48
Chapter 3.3 --- Distributive modules over arithematical rings --- p.52
Chapter 4 --- Chinese Modules and Universal Chinese rings --- p.59
Chapter 4.1 --- Introduction --- p.59
Chapter 4.2 --- Chinese Modules and CRT modules --- p.61
Chapter 4.3 --- Universal Chinese Rings --- p.65
Chapter 4.4 --- Chinese modules over Noetherian domains --- p.70
Chapter 4.5 --- Remarks on CRT modules --- p.77
Bibliography --- p.80
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Sithole, Maria Lindiwe. "Frames of ideals of commutative f-rings". Diss., 2018. http://hdl.handle.net/10500/25328.
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In his study of spectra of f-rings via pointfree topology, Banaschewski [6] considers lattices of l-ideals, radical l-ideals, and saturated l-ideals of a given f-ring A. In each case he shows that the lattice of each of these kinds of ideals is a coherent frame. This means that it is compact, generated by its compact elements, and the meet of any two compact elements is compact. This will form the basis of our main goal to show that the lattice-ordered rings studied in [6] are coherent frames. We conclude the dissertation by revisiting the d-elements of Mart nez and Zenk [30], and characterise them analogously to d-ideals in commutative rings. We extend these characterisa-tions to algebraic frames with FIP. Of necessity, this will require that we reappraise a great deal of Banaschewski's work on pointfree spectra, and that of Mart nez and Zenk on algebraic frames.Mathematical Sciences
M. Sc. (Mathematics)
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"Generalized injectivity of non-commutative ring theory". Chinese University of Hong Kong, 1994. http://library.cuhk.edu.hk/record=b5895452.
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by Leung Yiu-chung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1994.
Includes bibliographical references (leaves 79-81).
Introduction
Chapter 1 --- Preliminaries --- p.1
Chapter 1.1 --- Chain Conditions --- p.4
Chapter 1.2 --- Categories of Modules --- p.5
Chapter 1.3 --- Projectivity and Injectivity --- p.5
Chapter 2 --- Generalization on CS modules --- p.11
Chapter 2.1 --- Introduction --- p.11
Chapter 2.2 --- Preliminaries --- p.12
Chapter 2.3 --- CS ring ´ؤ A generalization of injectivity --- p.16
Chapter 2.4 --- GCS ring ´ؤ A further generalization of CS ring --- p.19
Chapter 2.5 --- Generalized CS-modules --- p.24
Chapter 2.5.1 --- Direct Sum of Uniform Modules --- p.25
Chapter 2.5.2 --- GCS modules as direct sum of uniform modules --- p.28
Chapter 3 --- Ascending Chain Condition on Essential Submodules --- p.35
Chapter 3.1 --- Introduction --- p.35
Chapter 3.2 --- Preliminaries --- p.36
Chapter 3.3 --- Continuous rings with ACC on essential ideals --- p.38
Chapter 3.4 --- Analogous Results On CS-modules --- p.44
Chapter 3.5 --- Weak CS -modules --- p.50
Chapter 3.5.1 --- Decomposition of Weak CS-modules --- p.53
Chapter 3.6 --- Generalization of GCS-modules --- p.54
Chapter 3.7 --- On CESS-modules --- p.57
Chapter 3.7.1 --- On the decomposition of CESS-modules --- p.59
Chapter 4 --- Non-Singular Rings --- p.63
Chapter 4.1 --- CS-modules and CS-endomorphism rings --- p.63
Chapter 4.2 --- Categorical Equivalence and Morita Equivalence --- p.70
Chapter 4.3 --- Categories of CS-modules --- p.74
Bibliography --- p.79
38
Moore, William F. "Cohomology of products of local rings". 2008. http://proquest.umi.com/pqdweb?did=1555891251&sid=7&Fmt=2&clientId=14215&RQT=309&VName=PQD.
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Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008.Title from title screen (site viewed Oct. 31, 2008). PDF text: v, 54 p. : ill. ; 769 K. UMI publication number: AAT 3313102. Includes bibliographical references. Also available in microfilm and microfiche formats.
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Cazaran, Jilyana. "Artinian rings, finite principal ideal rings and algebraic error-correcting codes". Thesis, 1998. https://eprints.utas.edu.au/19055/1/whole_CazaranJilyana1998_thesis.pdf.
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This thesis contains structure theorems for several types of Artinian
rings, in particular, finite rings, commutative Artinian rings containing an
identity, finite commutative rings containing an identity, and semisimple Artinian
semigroup-graded rings. Chapter 1 provides an introduction to Artinian
rings, semigroup-graded rings and some algebraic coding theory. Except for
a small percentage of lemmas which are referenced, all the theory contained
in Chapters 2 to 5 is new. It is original work either by myself or in collaboration
with my supervisor Andrei V. Kelarev. Some results obtained while
conducting my Ph.D. research have appeared in [8] to [22].
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Francis, Maria. "Grobuer Basis Algorithms for Polynomial Ideal Theory over Noetherian Commutative Rings". Thesis, 2017. http://etd.iisc.ac.in/handle/2005/3543.
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One of the fundamental problems in commutative algebra and algebraic geometry is to understand the nature of the solution space of a system of multivariate polynomial equations over a field k, such as real or complex numbers. An important algorithmic tool in this study is the notion of Groebner bases (Buchberger (1965)). Given a system of polynomial equations, f1= 0,..., fm = 0, Groebner basis is a “canonical" generating set of the ideal generated by f1,...., fm, that can answer, constructively, many questions in computational ideal theory. It generalizes several concepts of univariate polynomials like resultants to the multivariate case, and answers decisively the ideal membership problem. The dimension of the solution set of an ideal I called the affine variety, an important concept in algebraic geometry, is equal to the Krull dimension of the corresponding coordinate ring, k[x1,...,xn]/I. Groebner bases were first introduced to compute k-vector space bases of k[x1,....,xn]/I and use that to characterize zero-dimensional solution sets. Since then, Groebner basis techniques have provided a generic algorithmic framework for computations in control theory, cryptography,
formal verification, robotics, etc, that involve multivariate polynomials over fields.
The main aim of this thesis is to study problems related to computational ideal theory over Noetherian commutative rings (e.g: the ring of integers, Z, the polynomial ring over a field, k[y1,…., ym], etc) using the theory of Groebner bases. These problems surface in many domains including lattice based cryptography, control systems, system-on-chip design, etc.
Although, formal and standard techniques are available for polynomial rings over fields, the presence of zero divisors and non units make developing similar techniques for polynomial rings over rings challenging.
Given a polynomial ring over a Noetherian commutative ring, A and an ideal I in A[x1,..., xn], the first fundamental problem that we study is whether the residue class polynomial ring, A[x1,..., xn]/I is a free A-module or not. Note that when A=k, the answer is always ‘yes’ and the k-vector space basis of k[x1,..., xn]/I plays an important role in computational ideal theory over fields. In our work, we give a Groebner basis characterization for A[x1,...,xn]/I to have a free A-module representation w.r.t. a monomial ordering. For such A-algebras, we give an algorithm to compute its A-module basis. This extends the Macaulay-Buchberger basis theorem to polynomial rings over Noetherian commutative rings. These results help us develop a theory of border bases in A[x1,...,xn] when the residue class polynomial ring is
finitely generated. The theory of border bases is handled as two separate cases: (i) A[x1,...,xn]/I is free and (ii) A[x1,...,xn]/I has torsion submodules.
For the special case of A = Z, we show how short reduced Groebner bases and the characterization for a free A-module representation help identify the cases
when Z[x1,...,xn]/I is isomorphic to ZN for some positive integer N. Ideals in such Z-algebras are called ideal lattices. These structures are interesting since this means we can use the algebraic structure, Z[x1,...,xn]/I as a representation for point lattices and extend all the computationally hard problems in point lattice theory to Z[x1,...,xn]/I . Univariate ideal lattices are widely used in lattice based cryptography for they are a more compact representation for lattices than matrices. In this thesis, we give a characterization for multivariate ideal lattices and construct collision resistant hash functions based on them using Groebner basis techniques. For the construction of hash functions, we define a worst case problem,
shortest substitution problem w.r.t. an ideal in Z[x1,...,xn], and establish hardness results for this problem.
Finally, we develop an approach to compute the Krull dimension of A[x1,...,xn]/I using Groebner bases, when A is a Noetherian integral domain. When A is a field, the Krull dimension of A[x1,...,xn]/I has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. We introduce the notion of combinatorial dimension of A[x1,...,xn]/I and give a Groebner basis method to compute it for residue class polynomial rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A[x1,...,xn]/I. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Groebner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.
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Francis, Maria. "Grobuer Basis Algorithms for Polynomial Ideal Theory over Noetherian Commutative Rings". Thesis, 2017. http://etd.iisc.ernet.in/2005/3543.
Pełny tekst źródłaStreszczenie:
One of the fundamental problems in commutative algebra and algebraic geometry is to understand the nature of the solution space of a system of multivariate polynomial equations over a field k, such as real or complex numbers. An important algorithmic tool in this study is the notion of Groebner bases (Buchberger (1965)). Given a system of polynomial equations, f1= 0,..., fm = 0, Groebner basis is a “canonical" generating set of the ideal generated by f1,...., fm, that can answer, constructively, many questions in computational ideal theory. It generalizes several concepts of univariate polynomials like resultants to the multivariate case, and answers decisively the ideal membership problem. The dimension of the solution set of an ideal I called the affine variety, an important concept in algebraic geometry, is equal to the Krull dimension of the corresponding coordinate ring, k[x1,...,xn]/I. Groebner bases were first introduced to compute k-vector space bases of k[x1,....,xn]/I and use that to characterize zero-dimensional solution sets. Since then, Groebner basis techniques have provided a generic algorithmic framework for computations in control theory, cryptography,
formal verification, robotics, etc, that involve multivariate polynomials over fields.
The main aim of this thesis is to study problems related to computational ideal theory over Noetherian commutative rings (e.g: the ring of integers, Z, the polynomial ring over a field, k[y1,…., ym], etc) using the theory of Groebner bases. These problems surface in many domains including lattice based cryptography, control systems, system-on-chip design, etc.
Although, formal and standard techniques are available for polynomial rings over fields, the presence of zero divisors and non units make developing similar techniques for polynomial rings over rings challenging.
Given a polynomial ring over a Noetherian commutative ring, A and an ideal I in A[x1,..., xn], the first fundamental problem that we study is whether the residue class polynomial ring, A[x1,..., xn]/I is a free A-module or not. Note that when A=k, the answer is always ‘yes’ and the k-vector space basis of k[x1,..., xn]/I plays an important role in computational ideal theory over fields. In our work, we give a Groebner basis characterization for A[x1,...,xn]/I to have a free A-module representation w.r.t. a monomial ordering. For such A-algebras, we give an algorithm to compute its A-module basis. This extends the Macaulay-Buchberger basis theorem to polynomial rings over Noetherian commutative rings. These results help us develop a theory of border bases in A[x1,...,xn] when the residue class polynomial ring is
finitely generated. The theory of border bases is handled as two separate cases: (i) A[x1,...,xn]/I is free and (ii) A[x1,...,xn]/I has torsion submodules.
For the special case of A = Z, we show how short reduced Groebner bases and the characterization for a free A-module representation help identify the cases
when Z[x1,...,xn]/I is isomorphic to ZN for some positive integer N. Ideals in such Z-algebras are called ideal lattices. These structures are interesting since this means we can use the algebraic structure, Z[x1,...,xn]/I as a representation for point lattices and extend all the computationally hard problems in point lattice theory to Z[x1,...,xn]/I . Univariate ideal lattices are widely used in lattice based cryptography for they are a more compact representation for lattices than matrices. In this thesis, we give a characterization for multivariate ideal lattices and construct collision resistant hash functions based on them using Groebner basis techniques. For the construction of hash functions, we define a worst case problem,
shortest substitution problem w.r.t. an ideal in Z[x1,...,xn], and establish hardness results for this problem.
Finally, we develop an approach to compute the Krull dimension of A[x1,...,xn]/I using Groebner bases, when A is a Noetherian integral domain. When A is a field, the Krull dimension of A[x1,...,xn]/I has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. We introduce the notion of combinatorial dimension of A[x1,...,xn]/I and give a Groebner basis method to compute it for residue class polynomial rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A[x1,...,xn]/I. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Groebner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.
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Uliczka, Jan. "Graded Rings and Hilbert Functions". Doctoral thesis, 2010. https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-201007066381.
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Die Arbeit basiert auf zwei Veröffentlichungen zur graduierten kommutativen Algebra: Thema des ersten Artikels ist die Übertragung eines klassischen Ergebnisses zur Höhe von Primidealen in Polynomringen auf allgemeine multigraduierte Ringe; einige Anwendungen für die multigraduierte Dimensionstheorie werden vorgestellt. Der zweite Artikel behandelt Hilbertreihen von Moduln über einem standard-graduierten Polynomring über einem Körper. Ausgehend von einem grundlegenden Ergebnis über gewisse formale Laurentreihen werden unter anderem die möglichen Hilbertreihen und h-Vektoren solcher Moduln charakterisiert.
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(6598226), Avram W. Steiner. "A-Hypergeometric Systems and D-Module Functors". Thesis, 2019.
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Let A be a d by n integer matrix. Gel'fand et al.\ proved that most A-hypergeometric systems have an interpretation as a Fourier–Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of A. A similar statement relating A-hypergeometric systems to exceptional direct images was proved by Reichelt. In the first part of this thesis, we consider a hybrid approach involving neighborhoods U of the torus of A and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated A-hypergeometric system is the inverse Fourier–Laplace transform of such a "mixed Gauss–Manin system".
If the semigroup ring of A is normal, we show that every A-hypergeometric system is "mixed Gauss–Manin".
In the second part of this thesis, we use our notion of mixed Gauss–Manin systems to show that the projection and restriction of a normal A-hypergeometric system to the coordinate subspace corresponding to a face are isomorphic up to cohomological shift; moreover, they are essentially hypergeometric. We also show that, if A is in addition homogeneous, the holonomic dual of an A-hypergeometric system is itself A-hypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.
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Grande, Vincent. "Exakte Moduln über dem von Manuel Köhler beschriebenen Ring". Masterarbeit, 2018. http://hdl.handle.net/11858/00-1735-0000-002E-E4FD-D.
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Nadareishvili, George. "A classification of localizing subcategories by relative homological algebra". Doctoral thesis, 2015. http://hdl.handle.net/11858/00-1735-0000-0028-867A-A.
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