Rozprawy doktorskie: „Units in rings and group rings”

1

Li, Yuanlin. "Units in integral group rings". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/nq23107.pdf.

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Ferguson, Ronald Aubrey. "Units in integral cyclic group rings for order L§RP§S". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq25045.pdf.

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3

Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras". Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368142.

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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

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4

Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras". Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1182/1/PhdThesisFaccinPaolo.pdf.

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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

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5

Silva, Renata Rodrigues Marcuz. "Unidades de ZC2p e Aplicações". Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-27062012-154612/.

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Seja p um número primo e seja uma raiz p - ésima primitiva da unidade. Considere os seguintes elementos i := 1 + + 2 + ... + i-1 para todo 1 i k do anel Z[] onde k = (p-1)/2. Nesta tese nós descrevemos explicitamente um conjunto gerador para o grupo das unidades do anel de grupo integral ZC2p; representado por U(ZC2p); onde C2p representa o grupo cíclico de ordem 2p e p satisfaz as seguintes condições: S := { -1, , u2, ... uk } gera U(Z[]) e U(Zp) = ou U(Zp)2 = e -1 U(Zp); que são verificadas para p = 7; 11; 13; 19; 23; 29; 53; 59; 61 e 67. Com o intuito de estender tais ideias encontramos um conjunto gerador para U(Z(C2p x C2) e U(Z(C2p x C2 x C2) onde p satisfaz as mesmas condições anteriores acrescidas de uma nova hipótese. Finalmente com o auxílio dos resultados anteriores apresentamos um conjunto gerador das unidades centrais do anel de grupo Z(Cp x Q8); onde Q8 representa o grupo dos quatérnios, ou seja, Q8 := .
Let p be an odd prime integer, be a pth primitive root of unity, Cn be the cyclic group of order n, and U(ZG) the units of the Integral Group Ring ZG: Consider ui := 1++2 +: : :+i1 for 2 i p + 1 2 : In our study we describe explicitly the generator set of U(ZC2p); where p is such that S := f1; ; u2; : : : ; up1 2 g generates U(Z[]) and U(Zp) is such that U(Zp) = 2 or U(Zp)2 = 2 and 1 =2 U(Zp)2; which occurs for p = 7; 11; 13; 19; 23; 29; 37; 53; 59; 61, and 67: For another values of p we don\'t know if such conditions hold. In addition, under suitable hypotheses, we extend these ideas and build a generator set of U(Z(C2p C2)) and U(Z(C2p C2 C2)): Besides that, using the previous results, we exhibit a generator set for the central units of the group ring Z(Cp Q8) where Q8 represents the quaternion group.

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Kitani, Patricia Massae. "Unidades de ZCpn". Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-26042012-235529/.

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Seja Cp um grupo cíclico de ordem p, onde p é um número primo tal que S = {1, , 1+\\theta, 1+\\theta+\\theta^2, · · · , 1 +\\theta + · · · + \\theta ^{p-3/2}} gera o grupo das unidades de Z[\\theta] e é uma raiz p-ésima primitiva da unidade sobre Q. No artigo \"Units of ZCp\" , Ferraz apresentou um modo simples de encontrar um conjunto de geradores independentes para o grupo das unidades do anel de grupo ZCp sobre os inteiros. Nós estendemos este resultado para ZCp^n , considerando que um conjunto similar a S gera o grupo das unidades de Z[\\theta]. Isto ocorre, por exemplo, quando \\phi(p^n)\\leq 66. Descrevemos o grupo das unidades de ZCp^n como o produto ±ker(\\pi_1) × Im(\\pi1), onde \\pi_1 é um homomorfismo de grupos. Além disso, explicitamos as bases de ker(\\pi_1) e Im(\\pi_1).
Let Cp be a cyclic group of order p, where p is a prime integer such that S = {1, , 1 + \\theta, 1 +\\theta +\\theta ^2 , · · · , 1 + \\theta + · · · +\\theta ^{p-3/2}} generates the group of units of Z[\\theta] and is a primitive pth root of 1 over Q. In the article \"Units of ZCp\" , Ferraz gave an easy way to nd a set of multiplicatively independent generators of the group of units of the integral group ring ZCp . We extended this result for ZCp^n , provided that a set similar to S generates the group of units of Z[\\theta]. This occurs, for example, when \\phi(p^n)\\leq 66. We described the group of units of ZCp^n as the product ±ker(\\pi_1) × Im(\\pi_1), where \\pi_1 is a group homomorphism. Moreover, we explicited a basis of ker(\\pi_1) and I m(\\pi_1).

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7

Stack, Cora. "Some results on the structure of the groups of units of finite completely primary rings and on the structure of finite dimensional nilpotent algebras". Thesis, University of Reading, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.262483.

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Filho, Antonio Calixto de Souza. "A importância das unidades centrais em anéis de grupo". Universidade de São Paulo, 2000. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-11122008-214317/.

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Na presente dissertação, discutimos o Problema do Isomorfismo em anéis de grupo para grupos infinitos da forma G × C, apresentado no artigo de Mazur [14], que enuncia um teorema mostrando a equivalência para o Problema do Isomorfismo entre essa classe de grupos infinitos e grupos finitos que satisfaçam a Conjectura do Normalizador. Nossa ênfase concentra-se na relação entre a Conjectura do Isomorfismo e a Conjectura do Normalizador, primeiramente, observada nesse artigo. Em seguida, consideramos um teorema de estrutura para as unidades centrais em anéis de grupo comunicado, pela primeira vez, no artigo de Jespers-Parmenter-Sehgal [9], e generalizado por Polcino Milies-Sehgal em [17], e Jespers-Juriaans em [7]. Evidenciamos a importância desse teorema para a Teoria de Anéis de Grupo e apresentamos uma nova demonstração para o teorema de equivalência de Mazur, considerando, para tanto, uma apropriada unidade central e sua estrutura, caracterizada pelo teorema comunicado para as unidades centrais. Concluímos a dissertação, descrevendo a construção do grupo das unidades centrais para o anel de grupo ZA5 , um grupo livre finitamente gerado de posto 1, utilizando a construção dada no artigo de Aleev [1].
In this dissertation, we discuss the Problem of the Isomorphism in group rings for infinite groups as G × C. This is presented in [14]. Such article states a theorem which shows an equivalence to the isomorphism problem between that infinite class group and finite groups verifying the Normalizer Conjecture. Our main purpose is the Normalizer Conjecture and the Isomorphism Conjecture relationship remarked in the cited article to the groups above. Following, we consider a group ring theorem to the central units subgroup firstly communicated in [9] and generalized in [17] and [7]. We point up the importance of such theorem to the Group Ring Theory and we give a short and a new demonstration to Mazurs equivalence theorem from using a suitable central unit altogether with its structure lightly by the Central Unit Theorem on focus. We conclude this work sketching the ZA5 central units subgroup on showing it is a free finitely generated group of rank 1 from the presenting construction in Aleevs article [1].

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9

Immormino, Nicholas A. "Clean Rings & Clean Group Rings". Bowling Green State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1374247918.

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10

Weber, Harald. "Group rings and twisted group rings for a series of p-groups". [S.l. : s.n.], 2003. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10761310.

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11

Turner, Emma Louise. "k-S-Rings". BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3670.

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For a finite group G we study certain rings called k-S-rings, one for each non-negative integer k, where the 1-S-ring is the centralizer ring of G. These rings have the property that the (k+1)-S-ring determines the k-S-ring. We show that the 4-S-ring determines G when G is any group with finite classes. We show that the 3-S-ring determines G for any finite group G, thus giving an answer to a question of Brauer. We show the 2-characters defined by Frobenius and the extended 2-characters of Ken Johnson are characters of representations of the 2-S-ring of G. We find the character table for the 2-S-ring of the dihedral groups of order 2n, n odd, and classify groups with commutative 3-S-ring.

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12

Dexter, Cache Porter. "Schur Rings over Infinite Groups". BYU ScholarsArchive, 2019. https://scholarsarchive.byu.edu/etd/8831.

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A Schur ring is a subring of the group algebra with a basis that is formed by a partition of the group. These subrings were initially used to study finite permutation groups, and classifications of Schur rings over various finite groups have been studied. Here we investigate Schur rings over various infinite groups, including free groups. We classify Schur rings over the infinite cyclic group.

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13

Srivastava, Ashish K. "Rings Characterized by Properties of Direct Sums of Modules and on Rings Generated by Units". Ohio University / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1181845354.

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14

Strouthos, I. "Stably free modules over group rings". Thesis, University College London (University of London), 2011. http://discovery.ucl.ac.uk/1325632/.

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We study finitely generated stably free modules over group rings associated to particular groups, primarily using Milnor’s construction of projective modules over rings and Quillen’s Patching Theorem. For an odd prime number q, we study the stably free module category of the group ring of the dihedral group of order 2q over the ring of integer coefficient polynomials in one variable. We show that, over this group ring, every stably free module is free if every stably free module is free over a particular localisation of a cyclic algebra over the ring of algebraic integers corresponding to the prime q. Using this result, we show that every stably free module is free over the group ring of the dihedral group of order 6 over the ring of integer coefficient polynomials, and we later extend this result to all dihedral groups of order 2q. Furthermore, we consider a class of projective modules, which we regard as locally free, in a certain precise sense, and which are related to stably free modules under certain conditions. We show that, over the integral group ring of the direct product of a quaternion group of order 8 and the infinite cyclic group, there are infinitely many such locally free modules of rank 1. In addition, we show that, over the algebra of integer coefficient quaternions, every projective module is free. Finally this thesis includes a treatment of resolutions of indecomposable modules over the integral group ring of the dihedral group of order 6. By studying the irreducible integral representations of this dihedral group, we construct certain resolutions for which there is a notion of duality and diagonalisability in the constituent homomorphisms, via the syzygy operator.

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15

Welch, Amanda Renee. "Characterizing Zero Divisors of Group Rings". Thesis, Virginia Tech, 2015. http://hdl.handle.net/10919/52949.

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The Atiyah Conjecture originates from a paper written 40 years ago by Sir Michael Atiyah, a famous mathematician and Fields medalist. Since publication of the paper, mathematicians have been working to solve many questions related to the conjecture, but it is still open. The conjecture is about certain topological invariants attached to a group G. There are examples showing that the conjecture does not hold in general. These examples involve something like the lamplighter group. We are interested in looking at examples where this is not the case. We are interested in the specific case where G is a finitely generated group in which the Pr'ufer group can be embedded as the center. The Pr'ufer group is a p-group for some prime p and its finite subgroups have unbounded order, in particular the finite subgroups of G will have unbounded order. To understand whether any form of the Atiyah conjecture is true for G, it will first help to determine whether the group ring kG of the group G has a classical ring of quotients for some field k. To determine this we will need to know the zero divisors for the group ring kG. Our investigations will be divided into two cases, namely when the characteristic of the field k is the same as the prime p for the Pr'ufer group and when it is different.
Master of Science

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16

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

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

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17

Kahn, Eric B. "THE GENERALIZED BURNSIDE AND REPRESENTATION RINGS". UKnowledge, 2009. http://uknowledge.uky.edu/gradschool_diss/707.

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Making use of linear and homological algebra techniques we study the linearization map between the generalized Burnside and rational representation rings of a group G. For groups G and H, the generalized Burnside ring is the Grothendieck construction of the semiring of G × H-sets with a free H-action. The generalized representation ring is the Grothendieck construction of the semiring of rational G×H-modules that are free as rational H-modules. The canonical map between these two rings mapping the isomorphism class of a G-set X to the class of its permutation module is known as the linearization map. For p a prime number and H the unique group of order p, we describe the generators of the kernel of this map in the cases where G is an elementary abelian p-group or a cyclic p-group. In addition we introduce the methods needed to study the Bredon homology theory of a G-CW-complex with coefficients coming from the classical Burnside ring.

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18

邵慰慈 i Wai-chee Shiu. "The algebraic structure and computation of Schur rings". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1992. http://hub.hku.hk/bib/B31233181.

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Shiu, Wai-chee. "The algebraic structure and computation of Schur rings /". [Hong Kong : University of Hong Kong], 1992. http://sunzi.lib.hku.hk/hkuto/record.jsp?B1329037X.

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20

Meyer, David Christopher. "Universal deformation rings and fusion". Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1883.

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This thesis is on the representation theory of finite groups. Specifically, it is about finding connections between fusion and universal deformation rings. Two elements of a subgroup N of a finite group Γ are said to be fused if they are conjugate in Γ, but not in N. The study of fusion arises in trying to relate the local structure of Γ (for example, its subgroups and their embeddings) to the global structure of Γ (for example, its normal subgroups, quotient groups, conjugacy classes). Fusion is also important to understand the representation theory of Γ (for example, through the formula for the induction of a character from N to Γ). Universal deformation rings of irreducible mod p representations of Γcan be viewed as providing a universal generalization of the Brauer character theory of these mod p representations of Γ. It is the aim of this thesis to connect fusion to this universal generalization by considering the case when Γ is an extension of a finite group G of order prime to p by an elementary abelian p-group N of rank 2. We obtain a complete answer in the case when G is a dihedral group, and we also consider the case when G is abelian. On the way, we compute for many absolutely irreducible FpΓ-modules V, the cohomology groups H2(Γ,HomFp(V,V) for i = 1, 2, and also the universal deformation rings R(Γ,V).

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21

Lee, Gregory Thomas. "Symmetric elements in group rings and related problems". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ59994.pdf.

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22

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

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23

Alahmadi, Adel Naif M. "Injectivity, Continuity, and CS Conditions on Group Rings". Ohio University / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1163521064.

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24

Mannan, W. H. "Low dimensional algebraic complexes over integral group rings". Thesis, University College London (University of London), 2007. http://discovery.ucl.ac.uk/1446153/.

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The realization problem asks: When does an algebraic complex arise, up to homotopy, from a geometric complex In the case of 2- dimensional algebraic complexes, this is equiv alent to the D2 problem, which asks when homological methods can distinguish between 2 and 3 dimensional complexes. We approach the realization problem (and hence the D2 problem) by classifying all pos sible algebraic 2- complexes and showing that they are realized. We show that if a dihedral group has order 2n, then the algebraic complexes over it are parametrized by their second homology groups, which we refer to as algebraic second homotopy groups. A cancellation theorem of Swan ( 11 ), then allows us to solve the realization problem for the group D$. Let X be a finite geometric 2- complex. Standard isomorphisms give 7r2(Ar) = H2(X Z), as modules over ni(X). Schanuel's lemma may then be used to show that the stable class of n2(X) is determined by k {X). We show how 7r3(X) maybe calculated similarly. Specif ically, we show that as a module over the fundamental group, (X) = S2{ir2{X)), where S2(ir2(X)) denotes the symmetric part of the module 7r2(X) z tt2(X). As a consequence, we are able to show that when the order of n (X) is odd, the stable class of 7r3(X) is also determined by ir {X). Given a closed, connected, orientable 5- dimensional manifold, with finite fundamen tal group, we may represent it, up to homotopy equivalence, by an algebraic complex. Poincare duality induces a homotopy equivalence between this algebraic complex and its dual. We consider how similar this homotopy equivalence may be made to the identity, (through appropriate choice of algebraic complex). We show that it can be taken to be the identity on 4 of the 6 terms of the chain complex. However, by finding a homological ob struction, we show that in general the homotopy equivalence may not be written as the identity.

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25

Kerby, Brent L. "Rational Schur Rings over Abelian Groups". BYU ScholarsArchive, 2008. https://scholarsarchive.byu.edu/etd/1491.

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In 1993, Muzychuk showed that the rational S-rings over a cyclic group Z_n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z_n. This idea is easily extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational S-rings over G in a natural way. Our main result is that any finite group may be represented as the automorphism group of such a rational S-ring over an abelian p-group. In order to show this, we first give a complete description of the automorphism classes and characteristic subgroups of finite abelian groups. We show that for a large class of abelian groups, including all those of odd order, the lattice of characteristic subgroups is distributive. We also prove a converse to the well-known result of Muzychuk that two S-rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic S-rings. Finally, we show that the automorphism group of any S-ring over a cyclic group is abelian.

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26

邵慰慈 i Wai-chee Shiu. "Schur rings over dihedral groups of order 2p". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1989. http://hub.hku.hk/bib/B31208873.

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Shiu, Wai-chee. "Schur rings over dihedral groups of order 2p /". [Hong Kong : University of Hong Kong], 1989. http://sunzi.lib.hku.hk/hkuto/record.jsp?B12364770.

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28

Eisele, Florian [Verfasser]. "Group rings over the p-Adic integers / Florian Eisele". Aachen : Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2012. http://d-nb.info/1022616773/34.

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29

Ahmed, Iftikhar. "Projective modules of group rings over quadratic number fields". Thesis, Durham University, 1994. http://etheses.dur.ac.uk/5669/.

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Let K be a quadratic number field, Ok its ring of integers, and G a cyclic group of order prime p. In this thesis, we study the kernel group D(O(_K)G) and obtain a number of results concerning its order and structure. For K imaginary, we also investigate the subset R(O(_k)G) of the locally free class group CI(O(_k)G) consisting of classes which occur as rings of integers of tame extensions of K with Galois group isomorphic to G. We calculate R(O(_k)G) under a variety of conditions and obtain, for an arbitrary tame extension L o( K with group G, invariants which determine the class of O(_L) in R(O(_k)G).

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30

DOROBISZ, KRZYSZTOF. "INVERSE PROBLEMS FOR UNIVERSAL DEFORMATION RINGS OF GROUP REPRESENTATIONS". Doctoral thesis, Università degli Studi di Milano, 2015. http://hdl.handle.net/2434/268872.

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We study representations of profinite groups over some particular type of local rings. More specifically, suppose a profinite group and its continuous finite dimensional representation over a finite field k are given. Then we are interested in studying all possibilities of lifting this representation to a representation over a ring that is complete, local, noetherian and whose residue field is isomorphic to k. For each problem of the above described type, an associated deformation functor can be defined. If such a functor is representable then the object representing it is called the universal deformation ring of the given representation. The following inverse problem is central in the thesis: which rings do occur as universal deformation rings in the introduced setting? The main results of the thesis go in two directions. Firstly, we show that every complete noetherian local commutative ring R with residue field k can be realized as a universal deformation ring of a continuous representation of a profinite group. This way we completely answer the stated question in its general form. Secondly, we address its modification and provide a non-trivial necessary condition for characteristic zero universal deformation rings of representations of groups that are finite.

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31

Juglal, Shaanraj. "Prime near-ring modules and their links with the generalised group near-ring". Thesis, Nelson Mandela Metropolitan University, 2007. http://hdl.handle.net/10948/714.

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In view of the facts that the definition of a ring led to the definition of a near- ring, the definition of a ring module led to the definition of a near-ring module, prime rings resulted in investigations with respect to primeness in near-rings, one is naturally inclined to attempt to define the notion of a group near-ring seeing that the group ring had already been defined and investigated into by, interalia, Groenewald in [7] . However, in trying to define the group near-ring along the same lines as the group ring was defined, it was found that the resulting multiplication was, in general, not associative in the near-ring case due to the lack of one distributive property. In 1976, Meldrum [19] achieved success in defining the group near-ring. How- ever, in his definition, only distributively generated near-rings were considered and the distributive generators played a vital role in the construction. In 1989, Le Riche, Meldrum and van der Walt [17], adopted a similar approach to that which led to a successful and fruitful definition of matrix near-rings, and defined the group near-ring in a more general sense. In particular, they defined R[G], the group near-ring of a group G over a near-ring R, as a subnear-ring of M(RG), the near-ring of all mappings of the group RG into itself. More recently, Groenewald and Lee [14], further generalised the definition of R[G] to R[S : M], the generalised semigroup near-ring of a semigroup S over any faithful R-module M. Again, the natural thing to do would be to extend the results obtained for R[G] to R[S : M], and this they achieved with much success.

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32

Semikina, Iuliia [Verfasser]. "G-theory of group rings for finite groups / Iuliia Semikina". Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1173789642/34.

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33

Tay, Julian Boon Kai. "Poincaré Polynomial of FJRW Rings and the Group-Weights Conjecture". BYU ScholarsArchive, 2013. https://scholarsarchive.byu.edu/etd/3604.

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FJRW-theory is a recent advancement in singularity theory arising from physics. The FJRW-theory is a graded vector space constructed from a quasihomogeneous weighted polynomial and symmetry group, but it has been conjectured that the theory only depends on the weights of the polynomial and the group. In this thesis, I prove this conjecture using Poincaré polynomials and Koszul complexes. By constructing the Koszul complex of the state space, we have found an expression for the Poincaré polynomial of the state space for a given polynomial and associated group. This Poincaré polynomial is defined over the representation ring of a group in order for us to take G-invariants. It turns out that the construction of the Koszul complex is independent of the choice of polynomial, which proves our conjecture that two different polynomials with the same weights will have isomorphic FJRW rings as long as the associated groups are the same.

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34

Renshaw, James Henry. "Flatness, extension and amalgamation in monoids, semigroups and rings". Thesis, University of St Andrews, 1986. http://hdl.handle.net/10023/11071.

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We begin our study of amalgamations by examining some ideas which are well-known for the category of R-modules. In particular we look at such notions as direct limits, pushouts, pullbacks, tensor products and flatness in the category of S-sets. Chapter II introduces the important concept of free extensions and uses this to describe the amalgamated free product. In Chapter III we define the extension property and the notion of purity. We show that many of the important notions in semigroup amalgams are intimately connected to these. In Section 2 we deduce that 'the extension property implies amalgamation' and more surprisingly that a semigroup U is an amalgamation base if and only if it has the extension property in every containing semigroup. Chapter IV revisits the idea of flatness and after some technical results we prove a result, similar to one for rings, on flat amalgams. In Chapter V we show that the results of Hall and Howie on perfect amalgams can be proved using the same techniques as those used in Chapters III and IV. We conclude the thesis with a look at the case of rings. We show that almost all of the results for semi group amalgams examined in the previous chapters, also hold for ring amalgams.

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35

Lännström, Daniel. "The structure of epsilon-strongly graded rings with applications to Leavitt path algebras and Cuntz-Pimsner rings". Licentiate thesis, Blekinge Tekniska Högskola, Institutionen för matematik och naturvetenskap, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-17809.

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The research field of graded ring theory is a rich area of mathematics with many connections to e.g. the field of operator algebras. In the last 15 years, algebraists and operator algebraists have defined algebraic analogues of important operator algebras. Some of those analogues are rings that come equipped with a group grading. We want to reach a better understanding of the graded structure of those analogue rings. Among group graded rings, the strongly graded rings stand out as being especially well-behaved. The development of the general theory of strongly graded rings was initiated by Dade in the 1980s and since then numerous structural results have been established for strongly graded rings. In this thesis, we study the class of epsilon-strongly graded rings which was recently introduced by Nystedt, Öinert and Pinedo. This class is a natural generalization of the well-studied class of unital strongly graded rings. Our aim is to lay the foundation for a general theory of epsilon-strongly graded rings generalizing the theory of strongly graded rings. This thesis is based on three articles. The first two articles mainly concern structural properties of epsilon-strongly graded rings. In the first article, we investigate a functorial construction called the induced quotient group grading. In the second article, using results from the first article, we generalize the Hilbert Basis Theorem for strongly graded rings to epsilon-strongly graded rings and apply it to Leavitt path algebras. In the third article, we study the graded structure of algebraic Cuntz-Pimsner rings. In particular, we obtain a partial classification of unital strongly, epsilon-strongly and nearly epsilon-strongly graded Cuntz-Pimsner rings up to graded isomorphism.

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36

Popov, Vladimir L., i vladimir@popov msk su. "Generators and Relations of the Affine Coordinate Rings of Connected". ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi972.ps.

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37

Pilewski, Nicholas J. "Units and Leavitt Path Algebras". Ohio University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1427464498.

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38

Grover, Parnesh Kumar Carleton University Dissertation Mathematics. "Orderings on division rings and normal subgroup structure of a unitary group". Ottawa, 1989.

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39

Nguyen, Long Pham Bao. "Fusion of Character Tables and Schur Rings of Dihedral Groups". BYU ScholarsArchive, 2008. https://scholarsarchive.byu.edu/etd/1429.

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A finite group H is said to fuse to a finite group G if the class algebra of G is isomorphic to an S-ring over H which is a subalgebra of the class algebra of H. We will also say that G fuses from H. In this case, the classes and characters of H can fuse to give the character table of G. We investigate the case where H is the dihedral group. In many cases, G can be completely determined. In general, G can be proven to have many interesting properties. The theory is developed in terms of S-ring of Schur and Wielandt.

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40

Helveston, John Knox. "Life rings a manual for developing small group ministry in an established church /". Theological Research Exchange Network (TREN), 1997. http://www.tren.com.

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41

Harris, Julianne S. "On the mod 2 general linear group homology of totally real number rings /". Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/5812.

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42

Szabo, Steve. "Convolutional Codes with Additional Structure and Block Codes over Galois Rings". Ohio University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1257792383.

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43

Köster, Iris [Verfasser], i Wolfgang [Akademischer Betreuer] Kimmerle. "Sylow numbers in character tables and integral group rings / Iris Köster ; Betreuer: Wolfgang Kimmerle". Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2017. http://d-nb.info/1130148572/34.

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44

Bächle, Andreas [Verfasser], i Wolfgang [Akademischer Betreuer] Kimmerle. "On torsion subgroups and their normalizers in integral group rings / Andreas Bächle. Betreuer: Wolfgang Kimmerle". Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2013. http://d-nb.info/1029460787/34.

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45

Long, Jane Holsapple. "The cohomology rings of the special affine group of Fp^2 and of PSL(3,p)". College Park, Md.: University of Maryland, 2008. http://hdl.handle.net/1903/8458.

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Thesis (Ph. D.) -- University of Maryland, College Park, 2008.
Thesis research directed by: Dept. of Mathematics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

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46

Gandhi, Raj. "Oriented Cohomology Rings of the Semisimple Linear Algebraic Groups of Ranks 1 and 2". Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42566.

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In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.

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47

Sommerhäuser, Yorck. "Yetter-Drinfel'd-Hopf algebras over groups of prime order /". Berlin [u.a.] : Springer, 2002. http://www.loc.gov/catdir/enhancements/fy0817/2002070799-d.html.

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48

Pitt, Melanie A. 1980. "Main group supramolecular coordination chemistry: Design strategies and dynamic assemblies". Thesis, University of Oregon, 2009. http://hdl.handle.net/1794/10287.

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xxi, 172 p. : ill. (some col.) A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
Main group supramolecular chemistry is a rapidly expanding field that combines the tools of coordination chemistry with the unusual and frequently unexpected coordination preferences exhibited by the main group elements. Application of established supramolecular design principles to those elements provides access to novel structure types and the possibility of new functionality introduced by the rich chemistry of the main group. Chapter I is a general review of the field of main group supramolecular chemistry, focusing in particular on the aspects of coordination chemistry and rational design strategies that have been thus far used to prepare polynuclear "metal"-ligand assemblies. Chapter II is a discussion of work toward supramolecular assemblies based on the coordination preferences of lead(II), in particular focusing on the 2-mercaptoacetamide and arylthiolate functionalities to target four-coordinate and three-coordinate geometries, respectively. Several possible avenues for further pursuing this research are suggested, with designs for ligands that may provide a more fruitful approach to the coordination of lead(II). Chapter III deals with the preparation of As 2 L 3 assemblies based on flexible ligand scaffolds. These assemblies exhibit structural changes in response to temperature and solvent, which may provide some insight into the subtle shape requirements involved in supramolecular guest binding. Chapter IV continues this work with an examination of how ligand structure affects mechanical coupling of stereochemistry between metal centers when the chelate ring is completed by a secondary bonding interaction such as the As-π contact. Finally, Chapter V presents a crystallographic and synthetic study of the nature of the interaction between pnictogens and arene rings. This interaction is ubiquitous in the coordination chemistry performed in the Johnson laboratory; understanding the role these interactions play in determining the final structure of supramolecular assemblies is vital to the preparation of more complex structures. Chapter VI presents a set of conclusions and outlook for future work on lead(II) supramolecular assemblies and the dynamic assemblies prepared from flexible organic scaffolds. This dissertation contains previously published and coauthored material.
Committee in charge: Kenneth Doxsee, Chairperson, Chemistry; Darren Johnson, Advisor, Chemistry; David Tyler, Member, Chemistry; Victoria DeRose, Member, Chemistry; Stephen Remington, Outside Member, Physics

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49

Iwaki, Edson Ryoji Okamoto. "Unidades Hipercentrais em Anéis de Grupo". Universidade de São Paulo, 2000. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-20052007-112821/.

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Grande parte dos problemas em Anéis de Grupo centraliza-se em torno do estudo do seu grupo de unidades. Torna-se importante então conhecermos a estrutura do grupo de unidades de um anel de grupo U(ZG). No entanto, salvo raras exceções, pouco se conhece acerca da estrutura de U(ZG). Uma das idéias para se conhecer um pouco mais sobre a estrutura do grupo de unidades seria estudarmos a sua série central superior. No caso em que o grupo G é finito, um resultado de Gruenberg pode ser usado para mostrar que a série central superior de U = U(ZG) estaciona. Este fato nos possibilita estudarmos o hipercentro de U(ZG). A fim de obter mais informações sobre as unidades hipercentrais de U(ZG), nós necessitamos de uma descrição dos subgrupos de torção do hipercentro de U(ZG), o qual obtemos através dos resultados de Bovdi sobre os subgrupos normais periódicos de U(ZG). De modo geral, utilizando os resultados de Bovdi obtemos uma classificação dos grupos periódicos G em função do subgrupo dos elementos % de torção do hipercentro de U(ZG). Neste momento, surgem algumas perguntas, as quais procuraremos expor neste trabalho. Entre elas: O limitante superior para a série central superior de U(ZG) depende do grupo G? Como determinar a altura central superior de U(ZG)? Neste momento é interessante salientarmos como a Conjectura do Normalizador nos possibilita obtermos uma estimativa para a altura central de U(ZG). Todas estas perguntas são respondidas no capítulo 4, como resultado dos trabalhos de Arora, Hales, Passi que nos garantem que neste caso a altura central de U(ZG) é no máximo 2. Embora a demonstração original deste fato, devido a Arora, Hales e Passi, não tenha utilizado a Conjectura do Normalizador, tomamos neste trabalho a idéia de supormos um provável caminho que levasse a este resultado obtendo estimativas para a altura central de U(ZG) utilizando a Conjectura do Normalizador e um teorema de Gross. Nosso intuito com isso foi o de conectarmos a resolução do problema em questão com um problema de pesquisa intensa atual na área, ou seja, a Conjectura do Normalizador. Nesse caso, surge mais uma pergunta: Quais os grupos G tais que U(ZG) admite altura central exatamente 0, 1 ou 2? Pergunta que é respondida por Arora, Hales e Passi também. Finalmente, mais um resultado de Arora, Hales e Passi nos mostram uma caracterização do hipercentro de U(ZG) que surpreendentemente bate com a estimativa dada pela Conjectura do Normalizador. É interessante notar aqui o aparecimento da Conjectura do Normalizador tanto para obtermos uma estimativa da altura central de U(ZG) como na caracterização do hipercentro de U(ZG). No capítulo 5 apresentamos a generalização dos resultados de Arora, Hales e Passi para o caso em que o grupo G é periódico, cujos resultados se devem basicamente a Y.Li. No caso em que o grupo G é periódico, Li mostrou que a altura central de U(ZG) é no máximo 2. E introduzindo o conceito de n-centro de um grupo, obtém-se uma caracterização do n-centro de U(ZG) em função dos resultados sobre o hipercentro do grupo de unidades.
A great deal of problems in Group Rings centralize around the study of its group of units. Hence it becomes important to know the structure of the group of units U(ZG). But with a few exceptions, we do not have much information about its structure. Trying to obtain more information about the structure of U(ZG), we could, for example, study the upper central series of U(ZG). In case G is finite, a result of Gruenberg implies that U(ZG) has finite central height. This fact allow us to study the hypercenter of U(ZG). In order to obtain more information about the hypercentral units of U(ZG) we need a description of the torsion subgroup of the hypercenter of U(ZG) which is provided by results of Bovdi on periodic normal subgroups of U(ZG). Gruenberg\'s result suscites some questions which we will try to answer in this work. Among them: The upper bound for the upper central serie of U(ZG) depends on of the group G? How could we determine the central height of U(ZG)? It is interesting to see how we could obtain an estimative for the central height of U(ZG) using the Normalizer Conjecture. All these questions are answered in chapter 4, as a consequence of Arora, Hales and Passi\'s work which guarantees us that in this case the central height of U(ZG) is at most 2. Nevertheless this result of Arora, Hales and Passi doesn\'t use the Normalizer Conjecture, we suppose here that the Normalizer Conjecture holds and used a result of Gross to obtain estimatives to the central height of U(ZG). Our aim was to connect the question discussed ahead with a intensive research problem, the Normalizer Conjecture. This arises the following question: For which groups does U(ZG) have central height exactly 0, 1 or 2? This question is also answered by Arora, Hales and Passi. Finally, another result of Arora, Hales and Passi present us a characterization of the hypercenter of U(ZG), which surprisingly satisfies the condition presented in the Normalizer Conjecture. It is interesting to observe here the appearing of Normalizer Conjecture to obtain an estimative for the central height of U(ZG) and to obtain a characterization of the hypercenter of U(ZG). In chapter 5 we present a result of Li which generalizes the result of Arora, Hales and Passi to the case when G is a periodic group. He proves that the central height of U(ZG) is also at most 2. Introducing the concept of n-center he was able to use the results about the hypercenter of U(ZG) to obtain a characterization of the n-center of U(ZG).

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50

Pallekonda, Seshendra. "Bounded category of an exact category". Diss., Online access via UMI:, 2008.

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