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Dissertations / Theses on the topic 'Noncommutative rings'
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1
Zhang, Yufei. "Orderings on noncommutative rings." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0013/NQ32804.pdf.
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Pandian, Ravi Samuel. "The structure of semisimple Artinian rings." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/2977.
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Proves two famous theorems attributed to J.H.M. Wedderburn, which concern the structure of noncommutative rings. The two theorems include, (1) how any semisimple Artinian ring is the direct sum of a finite number of simple rings; and, (2) the Wedderburn-Artin Theorem. Proofs in this paper follow those outlined in I.N. Herstein's monograph Noncommutative Rings with examples and details provided by the author.
3
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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Rennie, Adam Charles. "Noncommutative spin geometry." Title page, contents and introduction only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phr4163.pdf.
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Leroux, Christine M. "On universal localization of noncommutative Noetherian rings." Thesis, Northern Illinois University, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3567765.
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The concepts of middle annihilators and links between prime ideals have been useful in studying classical localization. Universal localization has given us an alternative to classical localization as an approach to studying the localization of noncommutative Noetherian rings at prime and semiprime ideals. There are two main ideas we explore in this thesis. The first idea is the relationship between certain middle annihilator ideals, links between prime ideals, and universal localization. The second idea is to explore the circumstances under which the universal localization of a ring will be Noetherian, in the case where the ring is finitely generated as a module over its center.
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Collier, Nicholas Richard. "On asymptotic stability of prime ideals in noncommutative rings." Thesis, University of Warwick, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403145.
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Brandl, Mary-Katherine. "Primitive and Poisson spectra of non-semisimple twists of polynomial algebras /." view abstract or download file of text, 2001. http://wwwlib.umi.com/cr/uoregon/fullcit?p3024507.
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Thesis (Ph. D.)--University of Oregon, 2001.Typescript. Includes vita and abstract. Includes bibliographical references (leaf 49). Also available for download via the World Wide Web; free to University of Oregon users.
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Low, Gordan MacLaren. "Injective modules and representational repleteness." Thesis, University of Glasgow, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.319776.
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Brazfield, Christopher Jude. "Artin-Schelter regular algebras of global dimension 4 with two degree one generators /." view abstract or download file of text, 1999. http://wwwlib.umi.com/cr/uoregon/fullcit?p9947969.
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Thesis (Ph. D.)--University of Oregon, 1999.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 103-105). Also available for download via the World Wide Web; free to University of Oregon users. Address: http://wwwlib.umi.com/cr/uoregon/fullcit?p9947969.
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Rogozinnikov, Evgenii [Verfasser], and Anna [Akademischer Betreuer] Wienhard. "Symplectic groups over noncommutative rings and maximal representations / Evgenii Rogozinnikov ; Betreuer: Anna Wienhard." Heidelberg : Universitätsbibliothek Heidelberg, 2020. http://d-nb.info/1215758219/34.
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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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Alves, Sergio Mota. "PI equivalencia e não equivalencia de algebras." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306373.
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Orientador: Plamen Emilov KoshlukovTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisticas e Computação Cientifica
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Resumo: As álgebras verbalmente primas são bem conhecidas em característica 0, já sobre corpos de característica p > 2 pouco sabemos sobre elas. Nesse trabalho vamos discutir algumas diferenças entre estes dois casos de característica sobre corpos infinitos. Iniciamos mostrando que o Teorema do Produto Tensorial de Kemer e duas de suas conseqüências não podem ser transportados para corpos infinitos de característica positiva p > 2. Em seguida, discutiremos algumas propriedades envolvendo as álgebras Aa;b, a saber, mostraremos que as álgebras Aa;b e Ma+b(E) não são PI-equivalentes e que as álgebras Aa;a e Ma;a (E) não são PI-equivalentes, e apresentaremos um resultado que enfatiza a importância dos monômios na determinação do ideal das identidades das álgebras Zn £ Z2-graduadas Aa;b em característica positiva. Por ¯m, apresentaremos modelos genéricos e calcularemos a dimensão de Gelfand-Kirillov para as álgebras relativamente livres de posto m nas variedades determinadas pelas álgebras E E, Aa;b e Ma;a(E) E. Como conseqüência, obteremos a prova da não PI- equivalência entre álgebras importantes para PI-teoria em característica positiva
Abstract: The verbally prime algebras are well understood in characteristic 0 while over a field of characteristic p > 2 little is known about them. In this work we discuss some sharp di®erences between these two cases for the characteristic. First we show that the so-called Kemer's Tensor Product Theorem and two of its consequences cannot be extended for infnite fields of positive characteristic p > 2. Afterwards we prove that the algebras Aa;b and Ma+b(E) are not PI equivalent, while the algebras Aa;a and Ma;a(E) E are PI equivalent. Moreover we obtain a result showing the importance of the monomials in the Zn £ Z2-graded T-ideal of the algebra Aa;b. Finally, we exhibit constructions of generic models. By using these models we compute the Gelfand-Kirillov dimension of the relatively free algebras of rank m in the varieties generated by E E, Aa;b, and Ma;a(E)E. As consequence we obtain the PI non equivalence of important algebras for the PI theory in positive characteristic
Doutorado
Algebra
Doutor em Matemática
13
Fidelis, Marcello. "Identidades polinomiais em algebras T-primas." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306378.
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Orientador: Plamen Emilov KoshlukovTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho estudamos os produtos tensoriais de T-ideais T-primos sobre corpos infinitos. O comportamento destes produtos tensoriais sobre corpos de caracteristica zero foi descrito por Kemer. Primeiramente mostramos, usando os m'etodos introduzidos por Regev, que tal descri¸cao vale se nos restringirmos apenas aos polinomios multilineares. Num segundo momento, aplicando identidades graduadas, mostramos que o Teorema sobre o Produto Tensorial 'e falso para os T-ideais das 'algebras M1,1(E) e E E, onde E 'e a 'algebra de Grassmann com dimensao infinita; M1,1(E) consiste das matrizes 2 × 2 sobre E tendo somente elementos pares (i.e. centrais) de E na diagonal principal, e a outra diagonal consistindo de elementos 'impares (anticomutitativos) de E. Entao voltamos nossa atencao para outros produtos tensoriais e estudamos suas respectivas identidades graduadas. Obtivemos novas demonstracoes de alguns dos casos do Teorema sobre o Produto Tensorial de Kemer. Note que estas demonstracoes nao dependem da teoria sobre a estrutura dos T-ideais, mas sao "elementares". Finalmente, usando outra vez identidades polinomiais graduadas, mostramos que o Teorema sobre o Produto Tensorial nao 'e valido em mais um caso: quando o corpo base possui caracteristica positiva. Isto vem para mostrar novamente que a teoria sobre a estrutura dos T-ideais e, essencialmente, uma teoria sobre identidades polinomiais multilineares.
Abstract: In this work we study tensor products of T-prime T-ideals over infinite fields. The behaviour of these tensor products over a field of characteristic zero was described by Kemer. First we show, using methods due to Regev, that such a description holds if one restricts oneself to multilinear polynomials only. Second, applying graded polynomial identities, we prove that the Tensor Product Theorem fails for the T-ideals of the algebras M1,1(E) and E E where E is the infinite dimensional Grassmann algebra; M1,1(E) consists of the 2×2 matrices over E having even (i.e. central) elements of E in the main diagonal, and the other diagonal consisting of odd (anticommuting) elements of E. Then we pass to other tensor products and study the respective graded identities. We obtain new proofs of some cases of Kemer's Tensor Product Theorem. Note that these proofs do not depend on the structure theory of T-ideals but are "elementary" ones. Finally, using graded polynomial identities once again, we show that the Tensor Product Theorem fails in one more case when the base field is of positive characteristic. All this comes to show once more that the structure theory of T-ideals is essentially about the multilinear polynomial identities
Doutorado
Matematica
Doutor em Matemática
14
Machado, Gustavo Grings. "Álgebras com identidades polinomais e suas dimensões de Gelfand-Kirillow." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306380.
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Orientador: Plamen Emilov KoshlukovDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica
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Resumo: Neste trabalho estudamos álgebras com identidades polinomiais, focando-se no estudo de álgebras associativas unitárias finitamente geradas. Nosso objetivo é fazer uma demonstração alternativa da não PI-equivalência de álgebras utilizando um invariante conhecido como dimensão de Gelfand-Kirillov. Este invariante tem ganhado importância ultimamente, uma vez que ele é relativamente fácil de calcular e, de certa forma, é capaz de diferenciar o modo com que duas álgebras crescem. Começamos com as definições e resultados básicos de álgebras, álgebras graduadas, identidades polinomiais (graduadas), reduções de identidades polinomiais, etc. Em seguida apresentamos alguns resultados de álgebras com identidades polinomiais finitamente geradas, que permitem uma melhor compreensão dos conceitos de altura e de dimensão de Gelfand-Kirillov. Depois estudamos o Teorema do Produto Tensorial de Kemer (TPT), donde se conclui a PI-equivalência (multilinear) envolvendo álgebras importantes na teoria de PI-álgebras, as álgebras T-primas. Em particular, conclui-se a PI-equivalência sobre corpos de característica zero de M1;1(E) e EE, em que E é a álgebra de Grassmann de um espaço vetorial de base enumerável. Enfim, finalizamos mostrando a não PI-equivalência sobre corpos infinitos de característica positiva maior que dois de M1;1(E) e E E, utilizando-se da dimensão de Gelfand-Kirillov
Abstract: In this work we study algebras with polynomial identities, focusing on the study of finitely generated unitary associative algebras. Our goal is to give an alternative proof of non PI-equivalence of algebras using an invariant known as Gelfand-Kirillov dimension. This invariant has gained importance lately since in many cases it is relatively easy to calculate and, surprisingly, it is able to differentiate the growth of two algebras. We begin with definitions and basic results of algebras, graded algebras, (graded) polynomial identities, reduction of polynomial identities, etc. Afterwards we present some results concerning finitely generated algebras with polynomial identities, which give a better comprehension of the notions of height and Gelfand-Kirillov dimension. Later on we study the Kemer's Tensor Product Theorem (TPT), from which we conclude (multilinear) PI-equivalence involving important algebras in PI-theory, the so called T-prime algebras. In particular, we deduce the PI-equivalence of M1;1(E) and E E over fields of characteristic zero, where E is the infinite dimensional Grassman algebra. Finally, we prove the non PI-equivalence of M1;1(E) and E E over infinite fields of prime characteristic greater than two by means of Gelfand-Kirillov dimension
Mestrado
Algebra
Mestre em Matemática
15
Schwarz, João Fernando. "Invariantes de anéis de operadores diferenciais: racionalidade de Gellfand-Kirillov, categorias de módulos, aplicações." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-25112018-231341/.
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Esta tese aborda, como a despeito da rigidez da álgebra de Weyl An(k), suas subálgebras de invariantes possuem uma rica teoria de invariantes: do ponto de vista de estrutura, se fizermos um estudo de equivalência birracional dentro da filosofia de Gelfand-Kirillov, temos o Problema de Noether Não-Comutativo, sobre o qual obtemos vários novos resultados (Capítulo 4). Do ponto de vista de representações, obtemos que suas subálgebras de invariantes, em vários casos, herdam de maneira natural a estrutura de módulos de Gelfand-Tsetlin da álgebra de Weyl (Capítulo 5), assim como uma noção natural de módulos holonômicos (Capítulo 6). Analisaremos resultados similares para outras álgebras semelhantes a Álgebra de Weyl, como anéis de operadores diferenciais no toro e álgebras de Weyl generalizadas (Capítulos 2, 4 e 5). Como aplicações, temos uma Conjectura de Gelfand-Kirillov para subálgebras esféricas de Cherednik (Capítulo 4); para a Conjectura de Gelfand-Kirillov para várias álgebras de Galois (Capítulos 5 e 7); e o problema de realizar U(L), em que L é uma algebra de Lie simples de tipo B,C,D, como uma ordem de Galois generalizando o caso de gln (Capítulo 5). Um Capítulo sobre o Problema de Noether Quântico e um resumo do artigo de Futorny e Schwarz, \"Quantum Linear Galois Algebras\", encerram a tese.This thesis discussess how, given the rigidity results on the Weyl Algebra An(k), its invariant subrings can nonetheless have an interesting invariant theory: from the structural point of view, a birrational equivalence study under the Gelfand-Kirillov philosophy gives us the Noncommutative Noether Problem, of which we obtain many new results (Chapter 4). From the point of view of representations, we obtain that their invariant rings, in many cases, have a natural theory of Gelfand-Tsetlin modules just like the Weyl Algebra (Chapter 5), and a natural notion of holonomic modules (Chapter 6). We discuss analogues results for algebras which are similar to the Weyl Algebra, such as the ring of differential operators on the torus and the generalized Weyl algebras (Chapters 2,4,5). As applications, we have a Gelfand-Kirillov Conjecture for spherical subalgebras of Cherednik (Chapter 4); for the Gelfand-Kirillov Conjecture of many Galois algebras (Chapter 5 and 7); and the problem to give a Galois structure to the algebra U(L), where L is a simple Lie algebra of type B,C,D -generalizing the case A (Chapter 5). A chapter about the Quantum Noether Problem and a resume of the article Quantum Linear Galois Algebras\" ends the thesis.
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Angeltveit, Vigleik. "Noncommutative ring spectra." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/34549.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 87-91).
Let A be an Ax ring spectrum. We give an explicit construction of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. Using this construction we can then study how THH(A) varies over the moduli space of AO structures on A, a problem which seems largely intractable using strictly associative replacements of A. We study how topological Hochschild cohomology of any 2-periodic Morava K-theory varies over the moduli space of AO structures and show that in the generic case, when a certain matrix describing the multiplication is invertible, the result is the corresponding Morava E-theory. If this matrix is not invertible, the result is some extension of Morava E-theory, and exactly which extension we get depends on the AO structure. To make sense of our constructions, we first set up a general framework for enriching a subcategory of the category of noncommutative sets over a category C using products of the objects of a non-E operad P in C. By viewing the simplicial category as a subcategory of the category of noncommutative sets in two different ways, we obtain two generalizations of simplicial objects.
(cont.) For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects.
by Vigleik Angeltveit.
Ph.D.
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Mello, Thiago Castilho de 1984. "Identidades polinomiais em álgebras matriciais sobre a álgebra de Grassmann." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306366.
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Orientador: Plamen Emilov KochloukovTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Nesta tese estudamos a álgebra genérica de M1;1 em dois geradores sobre um corpo infinito de característica diferente de 2. Descrevemos o centro desta álgebra e provamos que este é a soma direta do corpo com um ideal nilpotente da álgebra. Como consequência mostramos que este centro contém elementos não escalares, respondendo a uma pergunta feita por Berele. Em característica zero, estudamos também as identidades polinomiais de tal álgebra genérica e exibimos uma base finita para seu T-ideal, utilizando a descrição do seu centro e os resultados de Popov sobre as identidades de M1;1 em característica zero. Segue que tal base é formada pelos polin^omios [x1; x2][x3; x4][x5; x6], [[x1; x2][x3; x4]; x5] e s4, a identidade polinomial standard de grau 4. Por fim, utilizando ideias e resultados de Nikolaev sobre as identidades em duas variáveis de M2(K) em característica zero, mostramos que todas as identidades polinomiais em duas variáveis de M1;1 são consequências das identidades [[x1; x2]2; x1] e [x1; x2]³
Abstract: In this thesis, we study the generic algebra of M1;1 in two generators over an infinite field of characteristic different from 2. We describe the centre of this algebra and prove that this centre is a direct sum of the field and a nilpotent ideal of the algebra. As a consequence, we show that such centre contains nonscalar elements and thus we answer a question posed by Berele. In characteristic zero we also study the identities of this generic algebra and find a finite basis for its ideal of identities using the description of its centre and the results of Popov, about the identities of M1;1 in characteristic zero. It follows that such a basis is formed by the polynomials [x1; x2][x3; x4][x5; x6], [[x1; x2][x3; x4]; x5] and by s4, the standard identity of degree four. Finally, using ideas and results of Nikolaev about the identities in two variables of M2(K) in characteristic zero, we show that the polynomial identities in two variables of M1;1 follow from [[x1; x2]2; x1] and [x1; x2]³
Doutorado
Matematica
Doutor em Matemática
18
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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Schwarz, Joao Fernando. "Problema de Noether não-comutativo." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-31032015-113754/.
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Neste trabalho, temos o objetivo de introduzir o Problema de Noether Clássico e sua versão não- comutativa introduzida por J. Alev e F. Dumas em [AD06]. Discutiremos os principais casos co- nhecidos nos quais os problemas têm solução positiva, observando um forte paralelo entre os casos comutativo e não-comutativo. Cobriremos os tópicos preliminares necessários para entendimento dos enunciados: álgebras de Weyl, anéis de operadores diferenciais, extensões de Ore, localização em domínios não-comutativos, e corpos de Weyl. No Capítulo 5 deste trabalho, o aluno apresenta duas contribuições originais, obtidas em colaboração com seu orientador V. Futorny e F. Eshmatov: o Teorema 5.5, que é um resultado folclórico sobre invariantes de ações livres de grupos finitos no anel de operadores diferenciais de variedades afins; e o Teorema 5.6, que até onde sabemos é iné- dito, sobre invariantes dos Corpos de Weyl sob a ação de grupos de pseudo-reflexão. Todo material algébrico preliminar para a demonstração destes dois teoremas é incluído no texto da dissertação: um básico de teoria de invariantes, vários resultados da teoria de grupos de pseudo-reflexão, alguns conceitos básicos de geometria algébrica e álgebra comutativa, e uma discussão detalhada do quo- ciente de variedades afins sob ação de grupos finitos.In this work we aim to introduce the Classical Noether´s Problem, and its noncommutative version introduced by J. Alev and F. Dumas in [AD06]. We discuss the most well known cases of positive solution of these problems, pointing out a strong similarity between the cases of positive solution for the classical and noncommutative versions of the Problem. We cover the preliminary topics to understand the statement and solutions of these problems: Weyl algebras, differential operators rings, Ore extensions, noncommutative localization, and Weyl Skew-Fields. In the Chapter 5 of this dissertation, the student shows two original contributions, obtained in collaboration with his advisor V. Futorny and F. Eshmatov: Theorem 5.5, a result belonging to the folklore of the area of differential operators, describing its invariants under the free action of a finite group on an affine variety; and Theorem 5.6, about the invariants of the Weyl skew-fields under the action of pseudo-reflection groups. As far as we know, this result is new. All preliminary algebraic facts to prove these two facts are included in the body of this text. It includes some basic facts on invariant theory, many results about pseudo-reflection groups, some basic concepts of algebraic geometry and commutative algebra, and a detailed discussion of the quotient of an affine variety under the action of a finite group.
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Zhao, Xiangui. "Groebner-Shirshov bases in some noncommutative algebras." London Mathematical Society, 2014. http://hdl.handle.net/1993/24315.
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Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings.
This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5).
In Chapter 1, we introduce the background and the goals of the thesis.
In Chapter 2, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We find lower and upper bounds of the Gelfand-Kirillov dimension of a differential difference algebra under some conditions. We also give examples to demonstrate that our bounds are sharp.
In Chapter 3, we generalize the Groebner-Shirshov basis theory to differential difference algebras with respect to any left admissible ordering and develop the Groebner-Shirshov basis theory of finitely generated free modules over differential difference algebras. By using the theory we develop, we present an algorithm to compute the Gelfand-Kirillov dimensions of finitely generated modules over differential difference algebras.
In Chapter 4, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Groebner-Shirshov bases in skew solvable polynomial rings over fields. Our algorithm can detect redundant reductions and therefore it is more efficient than the traditional Buchberger algorithm.
Finally, in Chapter 5, we summarize our results and propose possible future work.
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Zhou, Yiqiang. "Noncommutative Prüfer rings and some generalizations." Thesis, 1993. http://hdl.handle.net/2429/2140.
Full textAbstract:
Noncommutative Prfifer rings appear naturally when one wants to transfer the known results for rings which arise in algebraic geometry (such as Dedekind, Krull and Priifer, valuation rings ...) to noncommutative rings. We remove the left-right symmetry condition of the noncommutative Prfifer rings introduced by Alajbegovic and Dubrovin, and introduce three natural generalizations, semi-Prfifer rings, right w-semi-Prfifer rings, and right w-Prfifer rings. We study the relations between the four concepts, and present the various properties that characterize them. We formulate and prove the basic facts for those rings (decompositions of such rings; Morita invariants of these notions; relations with some other notions). A new module-theoretic characterization of semiprime right Goldie rings is achieved by using the newly-defined concept of strongly compressible modules. The result is used to provide new characterizations of semiprime Goldie (prime right Goldie, or prime Goldie) rings, and right w-semi-Prfifer (semi-Prfifer, right w-Prfifer,or Prfifer) rings. In particular, the characterization of semiprime Goldierings of Lopez-Permouth, Rizvi, and Yousif using weakly-injective modules is an easy corollary of our results. We also study modules over noncommutative Priifer rings. It is shown that a module over a noncommutative Prfiferring has projective dimension at most one if and only if it is the union of a well-ordered continuous chain of submodules with each factor of the chain a finitely presented cyclic module. The result is used to present a characterization of divisible modules with projective dimension at most one over noncommutative Priifer rings, which generalizes a known result of L.Fuchs.
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Bruce, Chris. "C*-algebras from actions of congruence monoids." Thesis, 2020. http://hdl.handle.net/1828/11689.
Full textAbstract:
We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic
considerations; namely, we generalize the construction of Cuntz, Deninger,
and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions
of congruence monoids on rings of algebraic integers in number fields. Our motivation
for considering actions of congruence monoids comes from class field theory and work
on Bost–Connes type systems. We give two presentations and a groupoid model for
these algebras, and establish a faithfulness criterion for their representations. We
then explicitly compute the primitive ideal space, give a semigroup crossed product
description of the boundary quotient, and prove that the construction is functorial
in the appropriate sense. These C*-algebras carry canonical time evolutions, so that
our construction also produces a new class of C*-dynamical systems. We classify the
KMS (equilibrium) states for this canonical time evolution, and show that there are
several phase transitions whose complexity depends on properties of a generalized
ideal class group. We compute the type of all high temperature KMS states, and
consider several related C*-dynamical systems.Graduate
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Härtel, Johannes. "Reduktionssysteme zur Berechnung einer Auflösung der orthogonalen freien Quantengruppen Ao(n)." Doctoral thesis, 2008. http://hdl.handle.net/11858/00-1735-0000-0006-B3A7-7.
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